Codeforces Round 573 (Div. 1) |
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Finished |

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number theory

probabilities

*3400

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F. Tokitsukaze and Powers

time limit per test

3 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputTokitsukaze 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 $$$p$$$.

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.

Examples

Input

1 2 1

Output

-1

Input

3 5 1

Output

2 4 3

Input

2 5 4

Output

2 3

Input

4 9 8

Output

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.

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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