
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 that \( m \) and \( n \) are integers, the point \( P(m, n^2) \) is on the graph of the function
\[ 
f(x) = (x^2 - 4)(x^2 + 12x + 32) + 4. 
\]
Find the number of points \( P \) that satisfy these conditions.

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