
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 moving particle starts at the point $(3,5)$ and moves randomly until it hits one of the coordinate axes for the first time. At any point $(a,b)$, the particle moves to either $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, with each new point chosen with probability $\frac{1}{3}$. Find the probability that the particle will first hit the coordinate axes precisely at $(0,0)$. If this probability is expressed as $\frac{p}{3^q}$, where $p$ and $q$ are positive integers and $p$ is not divisible by $3$, determine $p + q$.

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