
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]


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Let $a$ and $b$ be positive integers such that $\gcd(a,b)$ is divisible by exactly $10$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $35$ distinct primes. If $a$ has fewer distinct prime factors than $b$, determine the maximum number of distinct prime factors $a$ can have.

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