
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
Define the Fibonacci numbers via  $F_0=0$ ,  $f_1=1$ , and  $F_{n-1}+F_{n-2}$ .

Olivia flips two fair coins at the same time, repeatedly, until she has flipped a tails on both, not necessarily on the same throw. She records the number of pairs of flips  $c$  until this happens (not including the last pair, so if on the last flip both coins turned up tails  $c$  would be  $0$ ). What is the expected value  $F_c$ ?

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