
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
Find the number of ordered 7-tuples of positive integers \((a_1, a_2, a_3, a_4, a_5, a_6, a_7)\) that satisfy the following properties:
(i) \(a_n + a_{n+1} = a_{n+2}\) for \(1 \leq n \leq 5\), and
(ii) \(a_6 = 2005\).

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