
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  $\left(4,4\right)$  and moves until it hits one of the coordinate axes for the first time. When the particle is at the point  $\left(a,b\right)$ , it moves at random to one of the points  $\left(a-1,b\right)$ ,  $\left(a,b-1\right)$ , or  $\left(a-1,b-1\right)$ , each with probability  $\tfrac{1}{3}$ , independently of its previous moves. The probability that it will hit the coordinate axes at  $\left(0,0\right)$  is  $\tfrac{m}{3^n}$ , where  $m$  and  $n$  are positive integers, and  $m$  is not divisible by  $3$ . Find  $m+n$ .

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