
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 $(5,5)$ on a grid and randomly moves to one of the adjacent points $(a-1, b)$, $(a, b-1)$, or $(a-1, b-1)$ with equal probability of $\frac{1}{3}$. It stops when it first hits any of the coordinate axes. The task is to find the probability that the particle hits the coordinate axes exactly at $(0,0)$ and express this probability as $\frac{m}{3^n}$, where $m$ and $n$ are positive integers with $m$ not divisible by $3$. Find the sum $m+n$.

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