Split cat function

Split the cat function into two functions `index` to always show the current day and `show` to show previous days.

Start by locating the current `cat` function and its callers in the TypeScript source. Check how it selects the current and previous days, then split those responsibilities into `index` and `show`; verify both call paths still return the intended day’s cat photo.

索引モデルが issue の本文から書いたものです。