
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
The Fibonacci numbers are defined by \( F_{0}=0, F_{1}=1 \), and \( F_{n}=F_{n-1}+F_{n-2} \) for \( n \geq 2 \). There exist unique positive integers \( n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6} \) such that
\[
\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}.
\]

Find \( n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6} \).

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