How to set Cookies.path in different folder using asp.net core 6 web api?

I am trying to set refresh token in different path in cookies. so that it will not send back to the server for every client request.

the following code what I tried to set different path when creating refresh token

private void SetRefreshToken(RefreshToken newRefreshToken)
{
    var cookieOptions = new CookieOptions
    {
        HttpOnly = true,
        Path = "/user/",
        Expires = newRefreshToken.ExpiredTime
    };
    Response.Cookies.Append("refreshToken", newRefreshToken.Token, cookieOptions);

    userAuth.RefreshToken = newRefreshToken.Token;
    userAuth.TokenCreated = newRefreshToken.CreatedTime;
    userAuth.TokenExpired = newRefreshToken.ExpiredTime;
}

If I don't set any path, I can see it in the browser -> Inspect -> Application -> cookies

, I know this is default path. But if I set path, I cannot see the refresh token. it's totally empty