
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
Determine how many distinct rational numbers k exist such that |k| < 300 and the equation 3x^2 + kx + 18 = 0 has at least one integer solution for x, and k is a multiple of 3.

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