
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
Given $a_i \in \{-1, 0, 1\}$ for $0 \le i \le 17$, find the number of $18$-tuples $(a_0, a_1, \dots, a_{17})$ satisfying $a_0 + 2a_1 + 2^2a_2 + \cdots + 2^{17}a_{17} = 2^{10}$.

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