`stringifySearchWith` does not call `stringify` method on strings

Which project does this relate to?

Router

Describe the bug

When using stringifySearchWith as stringifySearch option on the router instance with just the first argument supplied, strings do not get passed to the stringify method if not given a 2nd undocumented (https://tanstack.com/router/latest/docs/framework/react/guide/custom-search-param-serialization) parse method. By default JSON.parse is getting passed in there which results in strings only containing numbers to be encoded as "123" instead of just 123. A few weeks ago I supplied a custom stringifySearchWith to simply support encoding Set instances as well. This changed the behavior of how strings are encoded which in the end led to numeric string params to be ignored completely as I had a Zod validator defined with .string().optional().catch(undefined). Removing the .catch(undefined) revealed that it was receiving a number instead of a string.

Your Example Website or App

https://stackblitz.com/edit/github-qmrnwt4n?file=src%2Fmain.tsx

Steps to Reproduce the Bug or Issue

1. Click "Add numeric search param"
2. It crashes
3. Remove stringifySearch from router and reload
4. Click "Add numeric search param" again
5. It works as expected

Expected behavior

Difficult to say. I saw that this 2nd parse argument was introduced in https://github.com/TanStack/router/issues/654 but I don't really understand the background. At least this 2nd parameter should get documented or if not given, stringify should still be called?

Screenshots or Videos

https://github.com/user-attachments/assets/ad46d899-d078-4ee2-96be-2a4c1e7d8c24

Platform

• OS: macOS
• Browser: Edge
• Version: 1.95.3

Additional context

No response

Start with src/main.tsx in the linked StackBlitz and inspect the router's stringifySearchWith/stringifySearch configuration, then compare it with the default router behavior. Confirm the intended handling of numeric strings and whether the missing parse argument should be documented or change the serialization behavior.