
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
If the solutions of the equation $(6-k)(9-k)x^2 - (117-15k)x + 54 = 0$ with respect to $x$ are all integers, then the number of integer values of $k$ that satisfy this condition is ____.

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