
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
Determine the sum of the indices (exponents) $n_1 + n_2 + \cdots + n_r$ where $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either 0, 1, or -1 such that\[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2009.\]

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