The problem is that GitHub does not allow an action to trigger another action. However, if you as a user (e.g. with SSH key) authorize the push, then the doc-building action will trigger. However, if you are not using a user authorization for pushing the docs, then the gh-pages page build (which is also an action) is not triggered, see GitHub Pages build not triggered with GITHUB_TOKEN · Issue #1177 · JuliaDocs/Documenter.jl · GitHub.
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