
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
Find the largest positive integer $n$ such that the system of equations
$$
(x+1)^{2}+y_{1}^{2}=(x+2)^{2}+y_{2}^{2}=\cdots=(x+k)^{2}+y_{k}^{2}=\cdots=(x+n)^{2}+y_{n}^{2}
$$
has integer solutions $\left(x, y_{1}, y_{2}, \cdots, y_{n}\right)$.

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