
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
For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^7$, $8^9$, and $k$, where $k$ can now include the prime factor 5, i.e., $k = 2^a3^b5^c$?

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