
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
Find all pairs of strictly positive integers \((n, k)\) such that:

$$
\left(2^n - 2^0\right)\left(2^n - 2^1\right)\left(2^n - 2^2\right) \dots \left(2^n - 2^{n-1}\right) = k!
$$

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