
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 $\left(a_{1}, a_{2}, \ldots, a_{100}\right)$ be a permutation of $(1, 2, \ldots, 100)$.

What is the maximum number of perfect squares that can exist among the numbers $a_{1}, a_{1}+a_{2}, a_{1}+a_{2}+a_{3}, \ldots, a_{1}+\ldots+a_{100}$?

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