
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
The "Two Mice Boring Wall" problem in the "Nine Chapters on the Mathematical Art" is a classic problem in Chinese mathematics: "Now there is a wall as thick as a thousand feet, and two mice are boring through it from opposite sides. The big mouse bores one foot per day, and the small mouse also bores one foot per day. The big mouse doubles its daily boring distance, and the small mouse halves its daily boring distance. If the wall is thick enough, and $S_n$ represents the sum of the distances bored by the two mice in the first $n$ days, find the value of $m$, and ask on which day they will meet and how much each will have bored?"

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