
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
King William is located at  $(1, 1)$  on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the  $x, y$  axes and  $x+y = 4$ , he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on  $x+y = 4$  can be expressed as  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by Andrew Wen*

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