
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  $a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8$  be real numbers which satisfy
\[ S_3=S_{11}=1, \quad S_7=S_{15}=-1, \quad\text{and}\quad
S_5 = S_9 = S_{13} = 0, \quad \text{where}\quad S_n = \sum_{\substack{1 \le i < j \le 8  i+j = n}} a_ia_j. \]
(For example,  $S_5 = a_1a_4 + a_2a_3$ .)
Assuming  $|a_1|=|a_2|=1$ , the maximum possible value of  $a_1^2 + a_2^2 + \dots + a_8^2$  can be written as  $a+\sqrt{b}$  for integers  $a$  and  $b$ . Compute  $a+b$ .

*Based on a proposal by Nathan Soedjak*

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