
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 \((a_{1}, a_{2}, \ldots, a_{8})\) be a permutation of \((1,2, \ldots, 8)\). Find, with proof, the maximum possible number of elements of the set
\[ \left\{a_{1}, a_{1}+a_{2}, \ldots, a_{1}+a_{2}+\cdots+a_{8}\right\} \]
that can be perfect squares.

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