How to : Set searchParams in actions and get same in load functions

[Devr-pro]

Hey there! I have this search functionality implementaion going on while doing it am encountering this little problem where I do set search params in actions through form but at the same time when I submit & want to retrieve the same search param in load function it returns null, Here's what I am doing currently:

//+page.server.ts

export const load = (async ({ locals, fetch,url }) => {
   console.log(url)

   let req = await fetch(`/api/data?query=${JSON.parse(url.searchParams.get('query') as string)}`);
   let res = await req.json();

   return {
   	payload: res
   };
}) satisfies PageServerLoad;

export const actions={
search: async({request,url})=>{

   	throw redirect(303,`/home?query=${(await request.formData()).get('input')}`)

   }
}

Above solution works but its clearly unnecessary and not practical .

Want to achieve this :

//+page.server.ts

export const load = (async ({ locals, fetch,url }) => {
	console.log(url)

	let req = await fetch(`/api/data?query=${JSON.parse(url.searchParams.get('query') as string)}`);
	let res = await req.json();

	return {
		payload: res
	};
}) satisfies PageServerLoad;

export const actions={
 search: async({request,url})=>{

  let id=(await request.formData()).get('input') as string

  url.searchParams.set('query','id')
	}
}

Thanks!