
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 \( a \) and \( b \) be positive integers such that \( ab + a - b - 5 = 0 \). If \( M = \max \left(a^2 + b^2\right) \), find the number of integer points \((x, y)\) satisfying \( 3x^2 + 2y^2 \leq M \).

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