
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
A coin-tossing game starts with the number \( n \). If a coin lands heads, 1 is subtracted from \( n \); if it lands tails, 2 is subtracted. Let \( E_n \) be the expected number of tosses before the number becomes zero or negative. If \( \lim _{n \rightarrow \infty}\left(E_{n}-a n-b\right)=0 \), find the pair \( (a, b) \).

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