
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
Compute the number of sequences of real numbers  $a_1, a_2, a_3, \dots, a_{16}$  satisfying the condition that for every positive integer  $n$ ,
\[
a_1^n + a_2^{2n} + \dots + a_{16}^{16n} = \left \{ \begin{array}{ll} 10^{n+1} + 10^n + 1 & \text{for even } n  10^n - 1 & \text{for odd } n \end{array} \right. .
\]*Proposed by Evan Chen*

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