
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 $(5,3)$ and moves until it reaches the point $(0,0)$ without hitting any other part of the coordinate axes (except at $(0,0)$ itself). At each point $(a,b)$, the particle can randomly move to $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with equal probability $\frac{1}{3}$. The probability that it will hit $(0,0)$ without touching any other part of the coordinate axes is $\frac{m}{3^n}$, where $m$ and $n$ are positive 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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