ci: sync next branch to the website repo

[siketyan]

Summary

Added a push trigger to the next branch to sync it to the website repository.
See also biomejs/website#2779

Test Plan

The next branch in the website repo will be updated once this PR was merged

Docs

N/A

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Latest commit: 7b73e40

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[ematipico]

Why the change? Wasn't it working?

[siketyan]

Though it is case-insensitve, my IDE complains it as the function is defined as toJSON: https://docs.github.com/en/actions/reference/workflows-and-actions/expressions#tojson. Not a critical thing at all.