
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  $F(0)=0$ ,  $F(1)=\frac32$ , and  $F(n)=\frac{5}{2}F(n-1)-F(n-2)$   for  $n\ge2$ .

Determine whether or not  $\displaystyle{\sum_{n=0}^{\infty}\,
    \frac{1}{F(2^n)}}$  is a rational number.

(Proposed by Gerhard Woeginger, Eindhoven University of Technology)


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