
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  $\alpha$  be the unique real root of the polynomial  $x^3-2x^2+x-1$ . It is known that  $1<\alpha<2$ . We define the sequence of polynomials  $\left\{{p_n(x)}\right\}_{n\ge0}$  by taking  $p_0(x)=x$  and setting
\begin{align*}
p_{n+1}(x)=(p_n(x))^2-\alpha
\end{align*}
How many distinct real roots does  $p_{10}(x)$  have?

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