
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

1. Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given a sequence of positive integers $\{a_{n}\}$ such that $a_{1} = a$, and for any positive integer $n$, the sequence satisfies the recursion
$$
a_{n+1} = a_{n} + 2 \left[\sqrt{a_{n}}\right].
$$
(1) If $a = 8$, find the smallest positive integer $n$ such that $a_{n}$ is a perfect square.

(2) If $a = 2017$, find the smallest positive integer $n$ such that $a_{n}$ is a perfect square.

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