
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  $n \in \mathbb{N}_{\geq 2}.$  For any real numbers  $a_1,a_2,...,a_n$  denote  $S_0=1$  and for  $1 \leq k \leq n$  denote  $$ S_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k} $$  Find the number of  $n-$ tuples  $(a_1,a_2,...a_n)$  such that   $$ (S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n. $$  

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