I am trying to set a path on a cookie using cookies(), but I am unable to set paths other than the path of the current request. Am I just not understanding how cookies work, or is this a bug?

// app/api/route.ts
// This will not set a cookie on the browser. Only works if I set path to '/api'
export async function GET() {
  cookies().set({
    name: 'cookie',
    value: 'monster',
    path: '/api/foo'
  });
  return NextResponse.json({ hello: 'world' });
}