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The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small.Read more","ref_id",{"_153":154,"_155":156,"_157":31},"pub_date","attribution","turn_index",3,"ref_type","search","ref_index","https://www.erdosproblems.com/forum/thread/873",{"_162":182,"_164":-5,"_165":204,"_167":205,"_101":206,"_170":207,"_172":187,"_174":-5,"_135":188,"_189":208,"_191":-5,"_88":192,"_193":-5,"_194":-5},{"_162":182,"_164":-5,"_165":183,"_167":184,"_101":185,"_170":186,"_172":187,"_174":-5,"_135":188,"_189":190,"_191":-5,"_88":192,"_193":-5,"_194":-5},{"_162":163,"_164":-5,"_165":166,"_167":168,"_101":169,"_170":171,"_172":173,"_174":-5,"_135":175,"_176":177,"_178":15},"matched_text"," ","prefix","start_idx",7752,"end_idx",7753,[],"refs",[],"alt","","prompt_text","sources_footnote","sources",[179],"has_images",{"_145":180,"_143":158,"_152":181},"\n\n Erdős Problem #873 - Discussion thread\n\n","Erdős Problems","citeturn810974view0",565,587,[158],[],"([Erdős Problems](https://www.erdosproblems.com/forum/thread/873))","grouped_webpages","items",[195],"fallback_items","done","error","style",{"_145":180,"_143":158,"_152":181,"_151":-5,"_147":-5,"_196":-5,"_197":198,"_170":199,"_200":-5,"_201":-5},"attribution_segments","supporting_websites",[],[202],"hue","attributions",{"_153":31,"_155":203,"_157":31},"view",321,343,[158],[],[209],{"_145":180,"_143":158,"_152":181,"_151":-5,"_147":-5,"_196":-5,"_197":210,"_170":211,"_200":-5,"_201":-5},[],[212],{"_153":31,"_155":203,"_157":31},"content_type","parts",[216],"I read the thread. The problem asks whether, for every \\(\\varepsilon>0\\), some fixed block length \\(k\\) forces \n\\[\nF(A,X,k)L\\ge [d_1,\\ldots,d_m]\n\\gg_m\n\\frac{(L/N)^m}{(LH/N^2)^{m(m-1)/2}}.\n\\]\nSince \\(L[y_1,y_2,y_3]\\gg \\frac{N^3}{\\Delta},\n\\]\nso\n\\[\n\\Delta\\gg \\frac{N^3}{X}.\n\\]\nFix \\(r 0, there is a k\\r\\nsuch that F(A, X, k) < Xεfor every increasing sequence A and all sufficiently large X. This\\r\\nnote does not solve that problem. It gives two partial bounds obtained by combining a\\r\\nVandermonde lcm inequality with gcd information inside compact blocks. The main estimates\\r\\nproved here are\\r\\nF(A, X, 11) ≪ X223/676 log X\\r\\nand\\r\\nF(A, X, 6) ≪ X437/1320 log X.\\r\\nBoth exponents are only slightly below 1/3. The point of the note is not that these exponents\\r\\nare likely to be close to the truth, but that the usual triple-block barrier can be passed by\\r\\nforcing many pairwise gcds inside near-critical blocks.\\r\\n1 Introduction\\r\\nThe problem studied here is Problem 873 on the Erd˝os Problems site. It asks how often\\r\\nconsecutive blocks in an arbitrary increasing sequence of positive integers can have small least\\r\\ncommon multiple. The known starting point is the Erd˝os-Szemer´edi estimate\\r\\nF(A, X, 3) ≪ X1/3log X,\\r\\nand examples show that this is sharp for triples.\\r\\nThe difficulty is that the sequence A is completely arbitrary. There is no density assumption,\\r\\nno regular divisibility pattern, and no useful distribution hypothesis. A good block may be rare\\r\\nfor one reason and frequent for another.\\r\\nThe argument below has a simple shape. First, the usual Vandermonde lower bound for\\r\\nlcms gives the triple estimate and also controls dyadic intervals away from the scale ai ≍ X1/2.\\r\\nSecond, inside a compact block at the critical scale, a small lcm forces many large pairwise\\r\\ngcds. If these gcds are placed on the edges of a graph, their overlaps are controlled by triple\\r\\nand quadruple common gcds. This gives span lower bounds, and span lower bounds become\\r\\ncounting bounds by the usual greedy selection argument.\\r\\nI have deliberately removed the more speculative normal-form and fixed-parameter counting\\r\\nmaterial from earlier drafts. The present version keeps only the estimates whose proofs are\\r\\nwritten out from the basic lcm and gcd inequalities. The paper should therefore be read as a\\r\\npartial-progress note, not as a claimed solution of the original Xε question.\\r\\nThroughout the paper, implied constants may depend on fixed block lengths, but never on\\r\\nA, X, or the dyadic scale N.\\r\\n12 Notation and two lcm inequalities\\r\\nFor k ≥ 1, define\\r\\nF(A, X, k) = #{i : lcm(ai, ai+1, . . . , ai+k−1) < X}.\\r\\nIf N is dyadic, write FN (A, X, k) for the same count restricted to starts satisfying N ≤ ai < 2N.\\r\\nThe first input is the standard Vandermonde lcm lower bound.\\r\\nLemma 2.1 (Vandermonde lcm bound). Let 1 ≤ b1 < · · · < bm and let L = lcm(b1, . . . , bm).\\r\\nThen\\r\\nL ≥ Q\\r\\nb1b2 · · · bm\\r\\n1≤r lcm(x1, x2, x3) ≥\\r\\nx1x2x3\\r\\n(x2 − x1)(x3 − x2)(x3 − x1)\\r\\n≫\\r\\nN3\\r\\nH3\\r\\n.\\r\\nRearranging gives the claim.\\r\\nCorollary 3.2 (Triple bound). For every increasing sequence A ⊆ N,\\r\\nF(A, X, 3) ≪ X1/3log X.\\r\\nConsequently F(A, X, k) ≪ X1/3log X for every fixed k ≥ 3.\\r\\nProof. In a fixed dyadic interval [N, 2N), select good triple starts greedily so that no two selected\\r\\nstarts overlap. The selected numerical intervals are disjoint and each has span ≫ NX−1/3 by\\r\\nLemma ??. Hence there are O(X1/3) selected starts, and restoring the bounded overlap factor\\r\\ngives the same bound for all starts in the interval. Since every entry in a good block is less than\\r\\nX, there are O(log X) relevant dyadic intervals.\\r\\nFor the later argument we need a four-block version that works better away from N ≍ X1/2.\\r\\nLemma 3.3 (Primal-dual span bounds). Let m ≥ 3. Suppose\\r\\nN ≤ x1 < · · · < xm < 2N, lcm(x1, . . . , xm) < X,\\r\\nand put H = xm − x1. Then\\r\\nH ≫m N\\r\\n2/(m−1)X−2/(m(m−1))\\r\\nand\\r\\nH ≫m N\\r\\n2(m−2)/(m−1)X−(m−2)/m.\\r\\nIn particular, for m = 4,\\r\\nFN (A, X, 4) ≪ 1 +\\r\\nN\\r\\nmax(N2/3X−1/6, N4/3X−1/2)\\r\\n.\\r\\nThus\\r\\nFN (A, X, 4) ≪ 1 + min(X1/6N\\r\\n1/3\\r\\n, X1/2N\\r\\n−1/3\\r\\n).\\r\\nProof. The first bound follows directly from Lemma ??:\\r\\nX ≫m\\r\\nNm\\r\\nHm(m−1)/2\\r\\n.\\r\\nFor the dual bound, let L = lcm(x1, . . . , xm) and set dj = L/xj . The dj are distinct positive\\r\\ndivisors of L, and\\r\\n|dr − ds| = L\\r\\n|xs − xr|\\r\\nxrxs\\r\\n≪\\r\\nLH\\r\\nN2\\r\\n.\\r\\n3By Lemma ?? applied to the dj ,\\r\\nX > L ≥ lcm(d1, . . . , dm) ≫m\\r\\n(L/N)\\r\\nm\\r\\n(LH/N2)m(m−1)/2\\r\\n.\\r\\nSince the exponent of L on the right is negative for m ≥ 3 and L < X, this gives\\r\\nX ≫m\\r\\nXm−m(m−1)/2Nm(m−1)−m\\r\\nHm(m−1)/2\\r\\n.\\r\\nRearranging gives the dual span bound. The counting estimate follows by the same greedy\\r\\nselection used in the proof of Corollary ??.\\r\\n4 A pairwise gcd lower bound inside compact quadruples\\r\\nThe next lemma is the main local input. A compact good quadruple cannot have all its pairwise\\r\\ngcds small.\\r\\nLemma 4.1 (Pairwise gcds in a compact quadruple). Suppose\\r\\nN ≤ y1 < y2 < y3 < y4 < 2N, lcm(y1, y2, y3, y4) < X,\\r\\nand put H = y4 − y1. Then for every pair r < s,\\r\\ngcd(yr, ys) ≫\\r\\nN8\\r\\nX2H11 .\\r\\nProof. Let L = lcm(y1, y2, y3, y4) and\\r\\n∆ = Y\\r\\n1≤r/... or https://docs.google.com/document/d//...). Do not pass local filesystem paths, Windows paths, gdrive:// URIs, or plain names.\nurl?: string | null,\n// Google Drive file ID only (for example `1abcDEF...`). Do not pass extra parameters.\nid?: string | null,\n// Export MIME type for a native Google Doc, Sheet, or Slide file. Common examples: application/pdf, application/vnd.openxmlformats-officedocument.wordprocessingml.document, application/vnd.openxmlformats-officedocument.spreadsheetml.sheet, application/vnd.openxmlformats-officedocument.presentationml.presentation, text/markdown, text/plain, text/csv.\nmime_type?: string, // default: application/pdf\n}) => any;\n// Download the content and title of a Google Drive file. 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Leave null to use the default raw export format.\nraw_export_mime_type?: string | null,\n}) => any;\n// Find the index range of an exact text match in a Google Doc.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/find_document_text_range = (_: {\n// Exact document text to match. Prefer this over raw indexes when possible.\ntext_to_find: string,\n// Google Docs URL in the format https://docs.google.com/document/d//... or a raw Google Docs document ID. If you only know the document title or title keywords, call `search_documents` first instead of asking the user for a URL. Do not pass document titles, Drive open?id links, app:// URLs, or /document/create.\ndocument_url?: string | null,\n// Raw Google Docs document ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. 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Do not pass a full URL here.\ndocument_id?: string | null,\n// When true, include deleted comments and deleted replies in the result.\ninclude_deleted?: boolean, // default: False\n// Maximum comment threads to return on this page. Use the response nextPageToken to continue.\npage_size?: number, // default: 100\n// Opaque nextPageToken from a previous get_document_comments response.\npage_token?: string | null,\n}) => any;\n// Resolve the paragraph range containing a given document index.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_document_paragraph_range = (_: {\n// A Google Docs document index that falls within the paragraph you want to resolve.\nindex_within: number,\n// Google Docs URL in the format https://docs.google.com/document/d//... or a raw Google Docs document ID. If you only know the document title or title keywords, call `search_documents` first instead of asking the user for a URL. Do not pass document titles, Drive open?id links, app:// URLs, or /document/create.\ndocument_url?: string | null,\n// Raw Google Docs document ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\ndocument_id?: string | null,\n// Optional Google Docs tab ID. Use this to target a specific tab in a tabbed document. Exclude to get all tabs.\ntab_id?: string | null,\n}) => any;\n// Return table structures and cell text from a Google Doc.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_document_tables = (_: {\n// Google Docs URL in the format https://docs.google.com/document/d//... or a raw Google Docs document ID. If you only know the document title or title keywords, call `search_documents` first instead of asking the user for a URL. Do not pass document titles, Drive open?id links, app:// URLs, or /document/create.\ndocument_url?: string | null,\n// Raw Google Docs document ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\ndocument_id?: string | null,\n// Optional Google Docs tab ID. Use this to target a specific tab in a tabbed document. Exclude to get all tabs.\ntab_id?: string | null,\n}) => any;\n// Return paragraph text with document indexes for a Google Doc.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_document_text = (_: {\n// Google Docs URL in the format https://docs.google.com/document/d//... or a raw Google Docs document ID. 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Raw file IDs are preferred; Drive/Docs/Sheets/Slides URLs are also accepted.\nfileId: string,\n// Google Drive API `acknowledgeAbuse` query parameter for downloading abusive media when applicable.\nacknowledgeAbuse?: boolean | null,\n// Google Drive API `supportsAllDrives` query parameter.\nsupportsAllDrives?: boolean | null, // default: True\n// Deprecated Google Drive API `supportsTeamDrives` query parameter.\nsupportsTeamDrives?: boolean | null,\n// Google Drive API `includePermissionsForView` query parameter. Only `published` is supported.\nincludePermissionsForView?: string | null,\n// Google Drive API `includeLabels` query parameter: comma-separated label IDs to include in `labelInfo`.\nincludeLabels?: string | null,\n// Google Drive API partial response `fields` selector for the file metadata.\nfields?: string, // default: id,name,mimeType,webViewLink,createdTime,modifiedTime,parents,shared,driveId,hasAugmentedPermissions,capabilities/canShare,permissions(type,role,emailAddress,domain,displayName,allowFileDiscovery)\n}) => any;\n// Get presentation metadata and slide content for a Google Slides deck.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_presentation = (_: {\n// Google Slides URL in the format https://docs.google.com/presentation/d//... or a raw presentation ID. If you only know the deck title or title keywords, call `search_presentations` first instead of asking the user for a URL.\npresentation_url?: string | null,\n// Raw Google Slides presentation ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\npresentation_id?: string | null,\n}) => any;\n// Read user comments and replies on a Google Slides deck for additional review context.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_presentation_comments = (_: {\n// Google Slides URL in the format https://docs.google.com/presentation/d//... or a raw presentation ID. If you only know the deck title or title keywords, call `search_presentations` first instead of asking the user for a URL.\npresentation_url?: string | null,\n// Raw Google Slides presentation ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\npresentation_id?: string | null,\n// When true, include deleted comments and deleted replies in the result.\ninclude_deleted?: boolean, // default: False\n// Maximum comment threads to return on this page. Use the response nextPageToken to continue.\npage_size?: number, // default: 100\n// Opaque nextPageToken from a previous get_presentation_comments response.\npage_token?: string | null,\n}) => any;\n// Return a compact slide outline for stable slide targeting.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_presentation_outline = (_: {\n// Google Slides URL in the format https://docs.google.com/presentation/d//... or a raw presentation ID. If you only know the deck title or title keywords, call `search_presentations` first instead of asking the user for a URL.\npresentation_url: string,\n}) => any;\n// Return Google Slides table structures with row and column coordinates preserved.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_presentation_tables = (_: {\n// Google Slides URL\npresentation_url: string,\n}) => any;\n// Return only text content to reduce payload size.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_presentation_text = (_: {\n// Google Slides URL in the format https://docs.google.com/presentation/d//... or a raw presentation ID. If you only know the deck title or title keywords, call `search_presentations` first instead of asking the user for a URL.\npresentation_url?: string | null,\n// Raw Google Slides presentation ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\npresentation_id?: string | null,\n}) => any;\n// Return the current Google Drive user's profile information. This action takes no parameters.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_profile = () => any;\n// Get a single slide by object ID.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_slide = (_: {\n// Google Slides `objectId` for the target slide. Use the slide resource object ID, not the slide number.\nslide_object_id: string,\n// Google Slides URL in the format https://docs.google.com/presentation/d//... or a raw presentation ID. If you only know the deck title or title keywords, call `search_presentations` first instead of asking the user for a URL.\npresentation_url?: string | null,\n// Raw Google Slides presentation ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\npresentation_id?: string | null,\n}) => any;\n// Return slide metadata plus an inline thumbnail image for visual layout questions.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_slide_thumbnail = (_: {\n// Slide object ID to render as a thumbnail image.\nslide_object_id: string,\n// Google Slides URL in the format https://docs.google.com/presentation/d//... or a raw presentation ID. If you only know the deck title or title keywords, call `search_presentations` first instead of asking the user for a URL.\npresentation_url?: string | null,\n// Raw Google Slides presentation ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\npresentation_id?: string | null,\n// Thumbnail size. Defaults to MEDIUM. Use LARGE only when fine layout details matter.\nthumbnail_size?: \"LARGE\" | \"MEDIUM\" | \"SMALL\", // default: MEDIUM\n}) => any;\n// Read cell data from one or more bounded spreadsheet ranges using the CellData shape.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_spreadsheet_cells = (_: {\n// One or more A1 ranges including the sheet name, e.g. ['Sheet1!A1:C20']. Keep each range within existing sheet bounds.\nranges: string[],\n// Google Sheets spreadsheet URL in the format https://docs.google.com/spreadsheets/d//... or a raw spreadsheet ID. If you only know the spreadsheet title or title keywords, call `search_spreadsheets` first instead of asking the user for a URL.\nspreadsheet_url?: string | null,\n// Raw Google Sheets spreadsheet ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\nspreadsheet_id?: string | null,\n// Raw Google Sheets CellData field mask fragment. Examples: 'formattedValue,effectiveValue' or 'formattedValue,userEnteredValue,effectiveFormat(textFormat,numberFormat)'. Default: 'userEnteredValue,userEnteredFormat'. Prefer this action over `get_spreadsheet_range` unless you only need the plain cell values; use this action for formatting, formulas, validation, notes, hyperlinks, and other cell metadata.\ncell_fields?: string | null,\n}) => any;\n// Read user comments and replies on a Google Sheets spreadsheet for additional review context.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_spreadsheet_comments = (_: {\n// Google Sheets spreadsheet URL in the format https://docs.google.com/spreadsheets/d//... or a raw spreadsheet ID. If you only know the spreadsheet title or title keywords, call `search_spreadsheets` first instead of asking the user for a URL.\nspreadsheet_url?: string | null,\n// Raw Google Sheets spreadsheet ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\nspreadsheet_id?: string | null,\n// When true, include deleted comments and deleted replies in the result.\ninclude_deleted?: boolean, // default: False\n// Maximum comment threads to return on this page. Use the response nextPageToken to continue.\npage_size?: number, // default: 100\n// Opaque nextPageToken from a previous get_spreadsheet_comments response.\npage_token?: string | null,\n}) => any;\n// Get metadata about a spreadsheet.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_spreadsheet_metadata = (_: {\n// Google Sheets spreadsheet URL in the format https://docs.google.com/spreadsheets/d//... or a raw spreadsheet ID. If you only know the spreadsheet title or title keywords, call `search_spreadsheets` first instead of asking the user for a URL.\nspreadsheet_url?: string | null,\n// Raw Google Sheets spreadsheet ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\nspreadsheet_id?: string | null,\n// When true, return only sheet properties and chart IDs/titles.\ncharts_only?: boolean, // default: False\n// When true, include per-sheet conditional formatting rules in the response.\ninclude_conditional_format_rules?: boolean, // default: False\n}) => any;\n// Read only the plain values from a range of cells within a spreadsheet.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/get_spreadsheet_range = (_: {\n// Sheet tab name only (no ! or coordinates). For A1 notation compatibility, quote names with spaces/punctuation (e.g. 'Q1 Plan'). If the name contains a single quote, escape it as two single quotes inside the quoted name (e.g. 'O''Reilly').\nsheet_name: string,\n// Cell range only (A1 or R1C1), e.g. A1:B10, A:Z, or 1:200. Do not include the sheet name here (pass it only in sheet_name), and do not pass duplicated prefixes like Sheet1!Sheet1!A1:B10. Keep the range within existing sheet bounds. Use this action only when you need the plain values of a range; use `get_spreadsheet_cells` when you need cell values together with formatting, formulas, validation, notes, hyperlinks, or other cell metadata.\nrange: string,\n// Google Sheets spreadsheet URL in the format https://docs.google.com/spreadsheets/d//... or a raw spreadsheet ID. If you only know the spreadsheet title or title keywords, call `search_spreadsheets` first instead of asking the user for a URL.\nspreadsheet_url?: string | null,\n// Raw Google Sheets spreadsheet ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\nspreadsheet_id?: string | null,\n// The option to render the values, e.g. 'FORMATTED_VALUE', 'UNFORMATTED_VALUE' or 'FORMULA'. Use null for default.\nvalue_render_option?: \"FORMATTED_VALUE\" | \"UNFORMATTED_VALUE\" | \"FORMULA\" | null,\n}) => any;\n// List shared drives accessible to the user. This action takes no parameters.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/list_drives = () => any;\n// List the items directly contained in a Google Drive folder. Accepted parameters are only `url` and `top_k`.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/list_folder = (_: {\n// Google Drive folder URL (for example https://drive.google.com/drive/folders/). Do not pass `root`, `my-drive`, raw folder names, or local filesystem paths.\nurl: string,\n// Maximum number of items to scan in the folder. Parameter name is `top_k`.\ntop_k?: number, // default: 100\n}) => any;\n// Return the most recently modified documents accessible to the user. Accepted parameters are only `top_k` and `require_viewed_by_user`. Set `require_viewed_by_user=True` to only return files the current user has viewed.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/recent_documents = (_: {\n// Number of recent files to return. Parameter name is `top_k`.\ntop_k: number,\n// When true, return only files viewed by the authenticated user.\nrequire_viewed_by_user?: boolean, // default: False\n}) => any;\n// Search Google Drive files by query and return basic details. Accepted parameters are only `query`, `topn`, `special_filter_query_str`, `best_effort_fetch`, `fetch_ttl`, and `require_viewed_by_user`. Use clear, specific keywords such as project names, collaborators, or file types. Example: ``\"design doc pptx\"``. When using query, each search query is an AND token match. Meaning, every token in the query is required to be present in order to match. - Search will return documents that contain all of the keywords in the query. - Therefore, queries should be short and keyword-focused (avoid long natural language). - If no results are found, try the following strategies: 1) Use different or related keywords. 2) Make the query more generic and simpler. - To improve recall, consider variants of your terms: abbreviations, synonyms, etc. - Previous search results can provide hints about useful variants of internal terms — use those to refine queries. Use `special_filter_query_str` when you need precise MIME-type or metadata filters. It uses Google Drive v3 search (the `q` parameter). - Supported time fields: `modifiedTime`, `createdTime`, `viewedByMeTime`, `sharedWithMeTime` (ISO 8601, e.g., '2025-09-03T00:00:00'). - People/ownership filters: `'me' in owners`, `'user@domain.com' in owners`, `'user@domain.com' in writers`, `'user@domain.com' in readers`, `sharedWithMe = true`. - Type filters: `mimeType = 'application/vnd.google-apps.document'` (Docs), `...spreadsheet` (Sheets), `...presentation` (Slides), and `mimeType != 'application/vnd.google-apps.folder'` to exclude folders. or mimeType = 'application/vnd.google-apps.folder' to select folders. Set `require_viewed_by_user=True` to restrict results to files the current user has viewed. Do not pass unsupported fields like `top_k`, `max_results`, `page_size`, `folder_url`, `query_type`, `user_message`, `recency_days`, `driveId`, or `include_shared_drives`.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/search = (_: {\n// Keyword query for Drive search. Use concise terms like project/file names. This may be empty only when `special_filter_query_str` is provided.\nquery: string,\n// Maximum results to return. Parameter name is `topn` (not `top_k`, `max_results`, or `page_size`).\ntopn?: number, // default: 20\n// Optional raw Google Drive API `q` filter expression for advanced filtering.\nspecial_filter_query_str?: string, // default: \n// When true, attempt to fetch text content for each result.\nbest_effort_fetch?: boolean, // default: False\n// Best-effort fetch timeout in seconds when best_effort_fetch=true.\nfetch_ttl?: number, // default: 15.0\n// When true, keep only files viewed by the authenticated user.\nrequire_viewed_by_user?: boolean, // default: False\n}) => any;\n// Search spreadsheet rows containing a query string and return matching rows as a Markdown table.\n// This action may fail because it needs an OAuth permission that was not requested when this connection was created. Reconnect to request the new permission.\ntype /Google Drive/link_696a85e0cb788191bbe8bde3d34d95d3/search_spreadsheet_rows = (_: {\n// Sheet tab name only (no ! or coordinates). For A1 notation compatibility, quote names with spaces/punctuation (e.g. 'Q1 Plan'). If the name contains a single quote, escape it as two single quotes inside the quoted name (e.g. 'O''Reilly').\nsheet_name: string,\n// Cell range to search (A1 or R1C1), e.g. A1:Z200. Do not include the sheet name here (passing Sheet1!A1:Z200 will fail because sheet_name is prepended automatically).\nrange: string,\n// String to search for in any cell within each row.\nquery: string,\n// Google Sheets spreadsheet URL in the format https://docs.google.com/spreadsheets/d//... or a raw spreadsheet ID. If you only know the spreadsheet title or title keywords, call `search_spreadsheets` first instead of asking the user for a URL.\nspreadsheet_url?: string | null,\n// Raw Google Sheets spreadsheet ID only (for example `1abcDEF...`). Use this when you already have the ID from a prior search result. Do not pass a full URL here.\nspreadsheet_id?: string | null,\n// 1-based column numbers to return. Use null to return all columns (up to max_columns).\ncolumn_numbers?: number[] | null,\n// When true, always include the header row as the first data row in the output.\ninclude_header_row?: boolean, // default: True\n// Maximum number of rows to return from the selected range. Default is 500.\nmax_rows?: number, // default: 500\n// Maximum number of columns to return when column_numbers is null. Default is 100.\nmax_columns?: number, // default: 100\n}) => any;","api_tool.list_resources",{},{"_217":218,"_219":-5,"_93":275},{"_213":237,"_238":239,"_240":-5,"_45":274},{"_99":270,"_101":271,"_105":272,"_107":108,"_109":110,"_111":112,"_113":114,"_115":116,"_117":118,"_119":120,"_121":108,"_122":108,"_123":124,"_130":273},[],[],[],"99d4a79c-3402-4a67-9a98-9e3416045752","{\"path\":\"\",\"only_tools\":true,\"refetch_tools\":true}",{},{"_217":242,"_219":306,"_93":307},1777460989.30407,{"_213":45,"_214":305},{"_280":281,"_282":283,"_284":173,"_128":129,"_130":285,"_115":116,"_117":118,"_119":120,"_286":60,"_56":287,"_109":288,"_111":112,"_113":114,"_121":108,"_122":108,"_123":124,"_289":290,"_291":277,"_292":15,"_293":294,"_295":60},"finished_text","Thought for 18m 28s","initial_text","Reasoning","summarization_headline","bbb21ef8-6155-4f06-95ef-855f26f91b4c","poll_suppress_reconnect_banner",[296],"1b68adcb-e7b2-4ca5-a117-202f2bc29371","finished_duration_sec",1108,"reasoning_start_time","pro_skipped","pro_progress",100,"is_finalizing",{"_135":297,"_88":298,"_299":300,"_301":302,"_303":304},"notification","requested","notification_channel_id","jawbone","notification_channel_name","Tasks","notification_priority",4,[173],"a8km123",{},"dd",{"_319":320,"_321":322},"disablePrefetch","shouldPrefetchAccount","shouldPrefetchUser","shouldPrefetchSystemHints","promoteCss","disableSSR","stripModulepreloadImports","statsigGateEvaluationsPromise",["P",318],"traceId","5316658755881081351","traceTime",1778103542311]

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