Cookies in Next 14

I'm fairly new to SSR and server components.
I am having trouble understanding cookies behavior with ssr and client in mind.
My goal: Set cookie on NextJS backend, be able to see it in browser and in client components.
My code:
import { cookies } from "next/headers";
import { NextResponse } from "next/server";

export const dynamic = "force-dynamic"; // defaults to auto

export async function GET(request: Request, response: Response) {
    const myCookies = cookies().getAll();
    console.log(myCookies);

    return NextResponse.json({ message: "API GET" });
}

export async function POST(request: Request) {
    const cookieOptions = {
        expires: new Date("2025-01-01"),
    };
    cookies().set("test", "test", cookieOptions);
    const myCookies = cookies().getAll();
    console.log(myCookies);

    return NextResponse.json({ message: "API POST" });
}

This is my route.ts file.

import { cookies } from "next/headers";
import React from "react";

const AccountPage = async () => {
    const test = await fetch("http://localhost:3003/api", {
        method: "POST",
        credentials: "include",
    });

    const res = await fetch("http://localhost:3003/api");
    const data = await res.json();

    const myCookies = cookies().getAll();
    console.log(myCookies);

    return (
        <div>
            AccountPage
            <div>{JSON.stringify(data)}</div>
        </div>
    );
};

export default AccountPage;

This is my server component.
I know I said my goal was to read my cookie in browser but as first step I wanted to read it in my server component.
When I print myCookies on my POST route, I can see the cookie, but on the get request or on the component I get an empty array.
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