
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  $x_1,x_2,x_3, \cdots$  be a sequence defined by the following recurrence relation:
\[ \begin{cases}x_{1}&= 4 x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} \]
The first few terms of the sequence are  $x_1=4,x_2=9,x_3=41 \cdots$ Find all pairs of positive integers  $\{a,b\}$  such that  $x_a x_b$  is a perfect square.

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