
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 begins its movement from $(3,3)$ and randomly moves to one of the adjacent points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with equal probability of $\frac{1}{3}$. This process continues until the particle reaches any point on the coordinate axes. Determine the probability that the particle first reaches the coordinate axes at $(0,0)$ and represent this probability in the form $\frac{m}{3^n}$ where $m$ and $n$ are integers, and $m$ is not divisible by $3$. Calculate $m+n$.

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