fix(web-search): bind continuations to the serving API key - #4387

Node.js 20 is deprecated. The following actions target Node.js 20 but are being forced to run on Node.js 24: actions/checkout@11bd71901bbe5b1630ceea73d27597364c9af683, actions/github-script@60a0d83039c74a4aee543508d2ffcb1c3799cdea. For more information see: https://github.blog/changelog/2025-09-19-deprecation-of-node-20-on-github-actions-runners/