
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
Rosencrantz and Guildenstern each start with \$2013 and are flipping a fair coin. When the coin comes up heads, Rosencrantz pays Guildenstern \$1, and when the coin comes up tails, Guildenstern pays Rosencrantz \$1. Let \( f(n) \) be the number of dollars Rosencrantz is ahead of his starting amount after \( n \) flips. Compute the expected value of \( \max \{f(0), f(1), f(2), \ldots, f(2013)\} \).

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