Tokitsukaze and Powers
Description
Tokitsukaze is playing a room escape game designed by SkywalkerT. In this game, she needs to find out hidden clues in the room to reveal a way to escape.
After a while, she realizes that the only way to run away is to open the digital door lock since she accidentally went into a secret compartment and found some clues, which can be interpreted as:
- Only when you enter $n$ possible different passwords can you open the door;
- Passwords must be integers ranged from $0$ to $(m - 1)$;
- A password cannot be $x$ ($0 \leq x < m$) if $x$ and $m$ are not coprime (i.e. $x$ and $m$ have some common divisor greater than $1$);
- A password cannot be $x$ ($0 \leq x < m$) if there exist non-negative integers $e$ and $k$ such that $p^e = k m + x$, where $p$ is a secret integer;
- Any integer that doesn't break the above rules can be a password;
- Several integers are hidden in the room, but only one of them can be $p$.
Fortunately, she finds that $n$ and $m$ are recorded in the lock. However, what makes Tokitsukaze frustrated is that she doesn't do well in math. Now that she has found an integer that is suspected to be $p$, she wants you to help her find out $n$ possible passwords, or determine the integer cannot be $p$.
The only line contains three integers $n$, $m$ and $p$ ($1 \leq n \leq 5 \times 10^5$, $1 \leq p < m \leq 10^{18}$).
It is guaranteed that $m$ is a positive integer power of a single prime number.
If the number of possible different passwords is less than $n$, print a single integer $-1$.
Otherwise, print $n$ distinct integers ranged from $0$ to $(m - 1)$ as passwords. You can print these integers in any order. Besides, if there are multiple solutions, print any.
Input
The only line contains three integers $n$, $m$ and $p$ ($1 \leq n \leq 5 \times 10^5$, $1 \leq p < m \leq 10^{18}$).
It is guaranteed that $m$ is a positive integer power of a single prime number.
Output
If the number of possible different passwords is less than $n$, print a single integer $-1$.
Otherwise, print $n$ distinct integers ranged from $0$ to $(m - 1)$ as passwords. You can print these integers in any order. Besides, if there are multiple solutions, print any.
Samples
1 2 1
-1
3 5 1
2 4 3
2 5 4
2 3
4 9 8
2 4 7 5
Note
In the first example, there is no possible password.
In each of the last three examples, the given integer $n$ equals to the number of possible different passwords for the given integers $m$ and $p$, so if the order of numbers in the output is ignored, the solution is unique as shown above.