
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
Let  $(F_n)_{n\in{N^*}}$  be the Fibonacci sequence defined by
 $F_1=1$ ,  $F_2=1$ ,  $F_{n+1}=F_n+F_{n-1}$  for every  $n\geq{2}$ . Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]

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