
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
Archit and Ayush are walking around on the set of points  $(x,y)$  for all integers  $-1\leq x,y\leq1$ . Archit starts at  $(1,1)$  and Ayush starts at  $(1,0)$ . Each second, they move to another point in the set chosen uniformly at random among the points with distance  $1$  away from them. If the probability that Archit goes to the point  $(0,0)$  strictly before Ayush does can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m,n$ , compute  $m+n$ .

*2020 CCA Math Bonanza Lightning Round #3.2*

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