
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
A particle starts at the point $(6, 6)$ and moves until it hits one of the coordinate axes for the first time. The particle's movement options from the point $(a, b)$ include moving to $(a-1, b)$, $(a, b-1)$, or $(a-1, b-1)$, each with equal probability of $\frac{1}{3}$. The probability that the particle hits the axes specifically at the point $(0,0)$ is given by $\frac{m}{3^n}$, where $m$ and $n$ are integers, and $m$ is not divisible by $3$. Determine $m + n$.

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