
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
An ant starts at the origin of the Cartesian coordinate plane. Each minute it moves randomly one unit in one of the directions up, down, left, or right, with all four directions being equally likely; its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point  $(x,y)$  such that  $|x|+|y|=3$ . The expected number of moves it makes before stopping can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $100m+n$ .

*Proposed by Yannick Yao*

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