CF2087H Nim with Special Numbers

Let's recall the rules of the game "Nim". The game is played by two players who take turns. They have a certain number of piles of stones (the piles can be of the same size or different sizes; the size is defined as the number of stones in a pile). On their turn, a player must choose any non-empty pile of stones and remove any number of stones from it (at least one). The player who cannot make a move loses. In this problem, a modified version of the game will be used. There is a special set of numbers $ S $ that prohibits certain moves. For each number $ s_i $ in this set, if the current size of the pile of stones is strictly greater than $ s_i $ , it cannot be made strictly less than $ s_i $ in one move. In other words, if the current size of the pile is $ x $ , and the player tries to make a move that results in a size of $ y $ , and $ x > s_i > y $ for some number $ s_i \in S $ , such a move is not allowed. Your task is as follows. Initially, the set $ S $ is empty. You need to process queries of three types: - add a certain number $ s_i $ to the set; - remove a certain number $ s_i $ from the set; - determine who will win if this modified version of Nim is played with piles of stones of sizes $ a_1, a_2, \dots, a_k $ . Assume that both players play optimally.

The first line contains a single integer $ q $ ( $ 1 \le q \le 3 \cdot 10^5 $ ) — the number of queries. Each query is given in one line in one of the following formats: - $ 1 $ $ s_i $ ( $ 1 \le s_i \le 3 \cdot 10^5 $ ) — if the number $ s_i $ is in the set $ S $ , remove it; otherwise, add it to $ S $ ; - $ 2 $ $ k_i $ $ a_{i,1} $ $ a_{i,2} $ ... $ a_{i,k_i} $ ( $ 1 \le k_i \le 3 $ ; $ 1 \le a_{i,j} \le 3 \cdot 10^5 $ ) — determine who will win the game if it uses $ k_i $ piles of stones with sizes $ a_{i,1}, a_{i,2}, \dots, a_{i,k_i} $ .

For each query of type $ 2 $ , output First if the player making the first move will win, or Second if they will lose.