
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 sequence is defined \(F_1 = F_2 = 1\) and \(F_n = F_{n-1} + F_{n-2}\) for \(n \geq 3\).

Suppose Fibonacci numbers \(F_a, F_b, F_c\) form a geometric sequence with a common ratio and are in increasing order. If \(a + b + c = 2010\), determine \(a\).

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