
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 $2005 = c_1 \cdot 3^{a_1} + c_2 \cdot 3^{a_2} + \ldots + c_n \cdot 3^{a_n}$, where $n$ is a positive integer, $a_1, a_2, \ldots, a_n$ are distinct natural numbers (including 0, with the convention that $3^0 = 1$), and each of $c_1, c_2, \ldots, c_n$ is equal to 1 or -1. Find the sum $a_1 + a_2 + \ldots + a_n$.

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