
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
We define the recursive polynomial  $T_n(x)$  as follows: $T_0(x)=1$  $T_1(x)=x$  $T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$   $\forall n \in \mathbb N$ .**a)** find  $T_2(x),T_3(x),T_4(x)$  and  $T_5(x)$ .**b)** find all the roots of the polynomial  $T_n(x)$   $\forall n \in \mathbb N$ .

*Proposed by Morteza Saghafian*

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