
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
Roger the ant is traveling on a coordinate plane, starting at \((0,0)\). Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal probability. He will continue to move until he reaches some point \(P\) for which he could have reached \(P\) more quickly had he taken a different route. For example, if he goes from \((0,0)\) to \((1,0)\) to \((1,1)\) to \((1,2)\) to \((0,2)\), he stops at \((0,2)\) because he could have gone from \((0,0)\) to \((0,1)\) to \((0,2)\) in only 2 seconds. The expected number of steps Roger takes before he stops can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100 a + b\).

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