E. Enegue 的神秘灯笼AI (CI翻译)（GLM 5.3 Flash）
题目描述
Enegue is enjoying life as he strolls down a mildly overgrown path through the forest. At times, he could catch glimpses of the waning moon through gaps in the canopy. As the he followed the winding path through the undergrowth, Enegue suddenly felt the presence of a being somewhere just beyond his vision. Pausing momentarily, Enegue heard the rustling of leaves somewhere in the distance. Then, $n$ lanterns appeared in a row in front of Enegue.
Enegue 正一边沿着一条穿过森林、略微杂草丛生的小路漫步，一边享受着生活。偶尔，他能透过树冠的缝隙瞥见残月。当他沿着蜿蜒的小路穿过灌木丛时，Enegue 突然感到有一个存在就在他视野之外。他稍作停顿，听见远处某处传来树叶的沙沙声。接着，$n$ 盏灯笼排成一排出现在 Enegue 面前。AI (CI翻译)（GLM 5.3 Flash）
"Who," a voice asked.
“谁，”一个声音问道。AI (CI翻译)（GLM 5.3 Flash）
"Enegue the eyqs," responded Enegue, "who are you?"
“Enegue的eyqs，”Enegue回应道，“你是谁？”AI (CI翻译)（GLM 5.3 Flash）
The voice simply replied "who." Then, with a bright flash, Enegue saw the silhouette of an owl before all the lanterns went dark. After a brief moment of darkness, $k$ of the lanterns turned on again, and the owl was nowhere to be seen or heard. Suddenly struck by inspiration, Enegue decides to make you guess which $k$ of the $n$ lanterns are shining.
那个声音只是回答：“谁。”接着，伴随一道亮光，Enegue 在所有灯笼熄灭前看到了一只猫头鹰的剪影。短暂黑暗之后，$k$ 盏灯笼再次亮起，而猫头鹰已不见踪影，也听不到声音。突然灵光一现，Enegue 决定让你猜 $n$ 盏灯笼中哪 $k$ 盏在发光。AI (CI翻译)（GLM 5.3 Flash）
Enegue has a row of $n$ lanterns, of which $k$ are on. He wants you to guess which $k$ of the $n$ lanterns are on, but he will only answer your questions in a very cryptic fashion. For each question, you are allowed to choose any subset $S$ of the $n$ lanterns. Enegue will then count the number of lanterns on in $S$. Let this number be $x$, but Enegue will not tell you $x$. Instead, he will tell you the number of composite divisors of $x$. (eg. the composite divisors of 12 are $\{4,6,12\}$, so if $x=12$, then Enegue will answer with 3).
Enegue 有一排共 $n$ 盏灯笼，其中 $k$ 盏亮着。他想让你猜出这 n$k$ 盏灯中究竟是哪 k$n$ 盏亮着，但他只会以一种非常隐秘的方式回答你的问题。每次提问，你可以选择这 n$S$ 盏灯中的任意一个子集 S$n$。Enegue 会统计 $S$ 中亮着的灯笼数量。设这个数量为 $x$，但 Enegue 不会告诉你 $x$ 是多少。相反，他会告诉你 $x$ 的合数约数的个数。(例如，12 的合数约数是 $\{4,6,12\}$，所以如果 $x=12$，Enegue 会回答 3。)AI (CI翻译)（DeepSeek V4 Flash）
Since Enegue is a cactus, he will get bored after $2\times10^5$ questions, so you'll need to guess the $k$ lanterns that are on before he gets bored.
由于 Enegue 是个仙人掌，他在 $2\times10^5$ 次询问后就会感到无聊，所以你需要在 Enegue 感到无聊之前猜出亮着的是哪 $k$ 盏灯笼。AI (CI翻译)（GLM 5.3 Flash）
输入格式
The initial line of input contains two space-separated integers $n$ and $k$ ($4\le k\le n\le 100$), where $n$ is the total number of lanterns, and $k$ is the number of lanterns that are on.
输入的第一行包含两个以空格分隔的整数 $n$ 和 $k$($4\le k\le n\le 100$)，其中 $n$ 是灯笼总数，$k$ 是亮着的灯笼数。AI (CI翻译)（GLM 5.3 Flash）
This is an interactive problem. Your program should use standard input and output to communicate with the judge's program.
这是一道交互题。你的程序应当使用标准输入输出来与评测程序进行交互。AI (CI翻译)（GLM 5.3 Flash）
You are allowed $2\times10^5$ queries. Each query should contain the character '?' and a string $S$ of length $n$, separated by one space. The string $S$ represents the chosen subset, where the $i$-th character of $S$ is a '1' if and only if the $i$-th lantern is in the subset.
你最多可以进行 $2\times10^5$ 次询问。每次询问应包含字符 '?' 和一个长度为 $n$ 的字符串 $S$，中间用一个空格分隔。字符串 $S$ 表示选出的子集，其中当且仅当第 $i$ 盏灯笼在子集中时，$S$ 的第 $i$ 个字符为 '1'。AI (CI翻译)（GLM 5.3 Flash）
The response to each query will be a single integer as described above. A response of $-1$ means the query is malformed or you have submitted more than $2\times10^5$ queries, in which case your program should exit immediately and get a "Wrong Answer".
对于每个查询，返回值将是如上所述的单个整数。返回 $-1$ 表示该查询格式错误，或者你已提交超过 $2\times10^5$ 次查询；在这种情况下，你的程序应立即退出并得到 “Wrong Answer”。AI (CI翻译)（GLM 5.3 Flash）
Once you are ready to guess the $k$ lanterns, output a single line starting with the character '!' and a string $T$ of length $n$, separated by one space. The string $T$ represents the subset of $k$ lanterns in the same format as the string $S$ above. You will get "Accepted" if the guess is correct.
当你准备好猜测 $k$ 盏灯笼时，输出一行，以字符 '!' 开头，后接一个长度为 $n$ 的字符串 $T$，两者用一个空格分隔。字符串 $T$ 表示 $k$ 盏灯笼的子集，格式与上面的字符串 $S$ 相同。如果猜测正确，你将得到 "Accepted"。AI (CI翻译)（GLM 5.3 Flash）
备注
The sample input and output only show how the interaction works. It will probably get wrong answer.
样例输入和输出仅展示交互是如何进行的。它很可能会得到错误答案。AI (CI翻译)（GLM 5.3 Flash）
样例
样例 1
9 5
0
0// output to show interaction
? 111111111
? 000001111
! 101100101B. 按位异或AI (CI翻译)（GLM 5.3 Flash）
题目描述
Zhong Ziqian got an integer array $a_1, a_2, \ldots, a_n$ and an integer $x$ as birthday presents.
Zhong Ziqian 收到了一个整数数组 $a_1, a_2, \ldots, a_n$ 和一个整数 $x$ 作为生日礼物。AI (CI翻译)（GLM 5.3 Flash）
Every day after that, he tried to find a non-empty subsequence of this array 1≤b1<b2<…<bk≤n$1 \leq b_1 \lt b_2 \lt \ldots \lt b_k \leq n$, such that for all pairs $(i, j)$ where 1≤i<j≤k$1 \leq i \lt j \leq k$, the inequality $a_{b_i} \oplus a_{b_j} \geq x$ held. Here, $\oplus$ is the bitwise exclusive-or operation.
在那之后的每一天，他都试图找到这个数组的一个非空子序列 1≤b1<b2<…<bk≤n$1 \leq b_1 \lt b_2 \lt \ldots \lt b_k \leq n$，使得对于所有满足 1≤i<j≤k$1 \leq i \lt j \leq k$ 的数对 $(i, j)$，不等式 $a_{b_i} \oplus a_{b_j} \geq x$ 都成立。这里，$\oplus$ 是按位异或运算。AI (CI翻译)（GLM 5.3 Flash）
Of course, every day he must find a different subsequence.
当然，他每天必须找到一个不同的子序列。AI (CI翻译)（GLM 5.3 Flash）
How many days can he do this without repeating himself? As this number may be very large, output it modulo $998\,244\,353$.
他能在不重复自己的情况下这样做多少天？由于这个数可能非常大，请输出它模 $998\,244\,353$ 的结果。AI (CI翻译)（GLM 5.3 Flash）
输入格式
The first line of the input contains two integers $n$ and $x$ ($1 \leq n \leq 300\,000$, $0 \leq x \leq 2^{60}-1$). Here, $n$ is the size of the array.
输入的第一行包含两个整数 $n$ 和 $x$($1 \leq n \leq 300\,000$，$0 \leq x \leq 2^{60}-1$)。其中，$n$ 是数组的大小。AI (CI翻译)（GLM 5.3 Flash）
The next line contains $n$ integers $a_1, a_2, \ldots, a_n$: the array itself ($0 \leq a_i \leq 2^{60}-1$).
下一行包含 $n$ 个整数 $a_1, a_2, \ldots, a_n$：即数组本身($0 \leq a_i \leq 2^{60}-1$)。AI (CI翻译)（GLM 5.3 Flash）
输出格式
Output one integer: the number of subsequences of Ziqian's array such that bitwise xor of every pair of elements is at least $x$, modulo $998\,244\,353$.
输出一个整数：Ziqian 的数组的子序列个数，使得每一对元素的按位异或都至少为 $x$，对 $998\,244\,353$ 取模。AI (CI翻译)（GLM 5.3 Flash）
备注
In the first example, all $2^3-1$ non-empty subsequences are suitable.
在第一个样例中，所有 $2^3-1$ 个非空子序列都是合适的。AI (CI翻译)（GLM 5.3 Flash）
in the second example, two non-empty subsequences are not suitable, it is $b = [1, 2]$ and $b = [1, 2, 3]$, that is because $a_1 \oplus a_2 = 0 \oplus 1 = 1$ which is smaller than $2$.
在第二个样例中，有两个非空子序列不合适，分别是 $b = [1, 2]$ 和 $b = [1, 2, 3]$，这是因为 $a_1 \oplus a_2 = 0 \oplus 1 = 1$ 小于 $2$。AI (CI翻译)（GLM 5.3 Flash）
In the third example, $b = [1], b = [2], b = [3], b = [2, 3]$ are suitable.
在第三个样例中，b=[1]、b=[2]、b=[3]、b=[2,3]$b = [1], b = [2], b = [3], b = [2, 3]$ 是合适的。AI (CI翻译)（GLM 5.3 Flash）
样例
样例 1
3 0
0 1 27样例 2
3 2
0 1 25样例 3
3 3
0 1 24样例 4
7 4
11 5 5 8 3 1 335