
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
Given the set $P = \{x \mid 1 \leq x \leq 8, x \in \mathbb{Z}\}$, the line $y = 2x + 1$ and the hyperbola $mx^2 - ny^2 = 1$ have exactly one intersection point, where $m, n \in P$. The number of hyperbolas satisfying the above condition is _______.

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