
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  $\{a_n\}_{n=1}^{\infty}$  and  $\{b_n\}_{n=1}^{\infty}$  be sequences of integers such that  $a_1 = 20$ ,  $b_1 = 15$ , and for  $n \ge 1$ , \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2,  b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let  $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$ . Determine the number of positive integer factors of  $G$ .

*Proposed by Michael Ren*

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