CF1336E2.Chiori and Doll Picking (hard version)

This is the hard version of the problem. The only difference between easy and hard versions is the constraint of $m$. You can make hacks only if both versions are solved.

Chiori loves dolls and now she is going to decorate her bedroom!

As a doll collector, Chiori has got $n$ dolls. The $i$-th doll has a non-negative integer value $a_i$ ($a_i \lt 2^m$, $m$ is given). Chiori wants to pick some (maybe zero) dolls for the decoration, so there are $2^n$ different picking ways.

Let $x$ be the bitwise-xor-sum of values of dolls Chiori picks (in case Chiori picks no dolls $x = 0$). The value of this picking way is equal to the number of $1$-bits in the binary representation of $x$. More formally, it is also equal to the number of indices $0 \leq i \lt m$, such that $\left\lfloor \frac{x}{2^i} \right\rfloor$ is odd.

Tell her the number of picking ways with value $i$ for each integer $i$ from $0$ to $m$. Due to the answers can be very huge, print them by modulo $998\,244\,353$.

这是该问题的困难版本。简单版本与困难版本的唯一区别在于 $m$ 的约束条件。只有当两个版本均被解决时，才允许进行 hack。

千织喜爱玩偶，现在她准备装饰自己的卧室！

作为一名玩偶收藏家，千织拥有 $n$ 个玩偶。第 $i$ 个玩偶具有一个非负整数值 $a_i$（满足 $a_i \lt 2^m$，其中 $m$ 已给出）。千织打算从中选出若干个（可能为零个）玩偶用于装饰，因此共有 $2^n$ 种不同的选取方式。

令 $x$ 表示千织所选玩偶之值的按位异或和（若未选任何玩偶，则 $x = 0$）。该选取方式的价值定义为 $x$ 的二进制表示中 $1$ 的个数。更形式化地，它也等于满足 $\left\lfloor \frac{x}{2^i} \right\rfloor$ 为奇数的所有下标 $0 \leq i \lt m$ 的个数。

请为每个整数 $i$（从 $0$ 到 $m$）输出价值恰好为 $i$ 的选取方式数目。由于答案可能非常大，请对 $998\,244\,353$ 取模后输出。

The first line contains two integers $n$ and $m$ ($1 \le n \le 2 \cdot 10^5$, $0 \le m \le 53$) — the number of dolls and the maximum value of the picking way.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \lt 2^m$) — the values of dolls.

第一行包含两个整数 $n$ 和 $m$（$1 \le n \le 2 \cdot 10^5$，$0 \le m \le 53$）—— 分别表示玩偶的数量以及选取方式的最大值。

第二行包含 $n$ 个整数 $a_1, a_2, \ldots, a_n$（$0 \le a_i \lt 2^m$）—— 表示各个玩偶的值。

Print $m+1$ integers $p_0, p_1, \ldots, p_m$ — $p_i$ is equal to the number of picking ways with value $i$ by modulo $998\,244\,353$.

输出 $m+1$ 个整数 $p_0, p_1, \ldots, p_m$ — 其中 $p_i$ 表示价值恰好为 $i$ 的选取方案数，对 $998\,244\,353$ 取模。