
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  $a_n$  be a sequence recursively defined by  $a_0 = 0, a_1 = 1$  and  $a_{n+2} = a_{n+1} + a_n$ . Calculate the sum of  $a_n\left( \frac 25\right)^n$  for all positive integers  $n$ . For what value of the base  $b$  we get the sum  $1$ ?

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