
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 the number of pairs  $(a, b)$  of positive integers with the property that the greatest common divisor of  $a$  and  $ b$  is equal to  $1\cdot 2 \cdot 3\cdot ... \cdot50$ , and the least common multiple of  $a$  and  $ b$  is  $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$ .

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