
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
Call a sequence of positive integers  $(a_n)_{n \ge 1}$  a "CGMO sequence" if  $(a_n)_{n \ge 1}$  strictly increases, and for all integers  $n \ge 2022$ ,  $a_n$  is the smallest integer such that there exists a non-empty subset of  $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$   $A_n$  where  $a_n \cdot \prod\limits_{a \in A_n} a$  is a perfect square.

Proof: there exists  $c_1, c_2 \in \mathbb{R}^{+}$  s.t. for any "CGMO sequence"  $(a_n)_{n \ge 1}$  , there is a positive integer  $N$  that satisfies any  $n \ge N$ ,  $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$ .

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