
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
Consider a sequence  $x_1,x_2,\cdots x_{12}$  of real numbers such that  $x_1=1$  and for  $n=1,2,\dots,10$  let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose  $x_n>0$  for  $n=1,2,\dots,11$  and  $x_{12}=0$ . Then the value of  $x_2$  can be written as  $\frac{\sqrt{a}+\sqrt{b}}{c}$  for positive integers  $a,b,c$  with  $a>b$  and no square dividing  $a$  or  $b$ . Find  $100a+10b+c$ .

*Proposed by Michael Kural*

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