
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 point $(5,5)$ and moves randomly until it first hits a coordinate axis. From point $(a,b)$, the particle moves to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with an equal probability of $\frac{1}{3}$. The movement is independent of previous moves. We want to calculate the probability that the particle will first hit the coordinate axes at $(0,0)$. Present your answer in the form $\frac{m}{3^n}$, where $m$ and $n$ are integers, and $m$ is not divisible by $3$. Find the value of $m+n$.

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