
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 play a game in which they repeatedly flip a fair coin. Let \( a_{1}=4 \), \( a_{2}=3 \), and \( a_{n}=a_{n-1}+a_{n-2} \) for all \( n \geq 3 \). On the \( n \)th flip, if the coin is heads, Rosencrantz pays Guildenstern \( a_{n} \) dollars, and, if the coin is tails, Guildenstern pays Rosencrantz \( a_{n} \) dollars. If play continues for 2010 turns, what is the probability that Rosencrantz ends up with more money than he started with?

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