
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Let $m$ and $n$ be any integers. Find a pair of integers $p$ and $q$ such that the numbers

$$
x = m^{3} + p m n^{2} \quad \text{and} \quad y = 3 m^{2} n + q n^{3}
$$

satisfy that $x^{2} - y^{2}$ is a cube. Then, find a pair of integers $p$ and $q$ such that $x^{2} - 3 y^{2}$ is a cube.

Present the answer in LaTex format: \boxed{Your answer}
user