Use a github action to build the website

[kmcnellis]

• Compiles the website directory on push to main branch
• Run the build script on PRs to test it works
• Delete the compiled "website/public" directory
• Avoids waiting for someone to manually run the build script in order to update the gh-pages branch.

[rasmi]

This is great, thanks Kiana! Some things to figure out, which I can handle this week:

• Currently, there is a separate GitHub action push-gh-pages.yml that actually deploys the site to the gh-pages branch and is triggered on any change to website/public. We can have this new action just commit the built site to website/public and keep the other action as-is to actually deploy the site, OR we can combine the two so that this action both builds the site and pushes to gh-pages.

• Whether or not we want to keep the built website/public folder is a separate decision, though I am partial to keeping it. Because of how the gh-pages branch is set up, there is no history there, so the only way to currently see the actual changes and historical state of the published site is in the website/public directory history (or otherwise by checking out the repo for a given commit and compiling locally). This kind of historical archive is nice but not a big deal to get rid of IMO since change history is at least described in website/src.

• We should add some sort of status check / branch protection to prevent people from modifying website/public / gh-pages (and perhaps restrict them to only be modified via GitHub action). See Add branch protection for gh-pages #3. Does not need to be addressed by this PR.

• I would like to add a README.md to the root repo or possibly the website repo to document how these actions work and how to update the site (and the broader docs). See Write docs on how to update the docs #54, also does not have to be addressed by this PR.

Thanks!