
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
Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly $8$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $32$ distinct primes.

If $a$ has fewer distinct prime factors than $b$, then $a$ has at most how many distinct prime factors?

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