XOR Sum

You have an array of $n$ $k$-bit numbers $a_1, a_2, \ldots, a_n$.

Calculate $\sum_{i=1}^{n} \sum_{j=i+1}^{n} (a_i \oplus a_j)^x$.

$a \oplus b$ is the bitwise exclusive OR of $a$ and $b$.

Since the answer can be very large, output it modulo $998\,244\,353$.

The first line contains three integers $n$, $k$, $x$ ($1 \leq n, k, n \cdot k \leq 300\,000$, $1 \leq x \leq 3$), the length of the array, the number of bits in each number, and the power applied to each XOR result.

Each of the next $n$ lines contains a string $s_0, s_1, \ldots, s_{k-1}$ made of '0' and '1'. The $i$-th string gives $a_i = \sum_{i=0}^{k-1} s_i \cdot 2^i$.

Print one number: the sum of the $x$-th powers of the XOR results over all pairs, modulo $998\,244\,353$.

In the first sample, the array is $[5, 4, 4]$, and the answer is $(5 \oplus 4) + (5 \oplus 4) + (4 \oplus 4) = 1 + 1 + 0 = 2$.

In the second sample, the array is $[61, 38]$, and the answer is $(61 \oplus 38)^3 = 27^3 = 19683$.