
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 equation $a \cdot b + 125 = 30 \cdot \text{lcm}(a, b) + 24 \cdot \text{gcd}(a, b) + a \mod b$, where $\text{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a, b)$ denotes their least common multiple, assuming $a \geq b$, calculate the number of ordered pairs $(a, b)$ of positive integers that satisfy this equation.

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