Based Zeros

Barbara has always known how to write integers in the decimal system (base ten), using the digits $0, 1, 2, \ldots, 9$. Recently she learned that for any integer base $b \ge 2$, she can also write integers in base $b$, using the symbols with values from $0$ to $b-1$ as digits.

Barbara's favorite digit is $0$. Luckily, it looks the same in all bases.

Today Barbara is playing with a positive integer $n$. She wonders in which bases the representation of $n$ contains the most zeros. Find all such bases.

The first line contains the number of test cases $t$ ($1 \le t \le 1000$). Each test case is a single line containing one integer $n$ ($2 \le n \le 10^{18}$).

For each test case, print two integers $k$ and $m$ on the first line. Here $k$ is the maximum number of zeros that the representation of $n$ can have in any base, and $m$ is the number of bases that reach it.

On the second line, print the $m$ bases $b_1, b_2, \ldots, b_m$ in increasing order ($2 \le b_1 < b_2 < \cdots < b_m \le n$).

The representations with the maximum number of zeros for the sample cases are:

• $11 = \mathtt{1011}_2 = \mathtt{102}_3 = \mathtt{10}_{11}$ (one zero)
• $1007 = \mathtt{1101022}_3 = \mathtt{1007}_{10}$ (two zeros)
• $239 = \mathtt{11101111}_2 = \mathtt{1035}_6 = \mathtt{10E}_{15} = \mathtt{10}_{239}$ (one zero)

In $239 = \mathtt{10E}_{15}$, $\mathtt{E}$ is a digit with value $14$.