
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
Define a sequence of polynomials  $F_n(x)$  by  $F_0(x)=0, F_1(x)=x-1$ , and for  $n\geq 1$ ,   $$ F_{n+1}(x)=2xF_n(x)-F_{n-1}(x)+2F_1(x). $$   For each  $n$ ,  $F_n(x)$  can be written in the form   $$ F_n(x)=c_nP_1(x)P_2(x)\cdots P_{g(n)}(x) $$   where  $c_n$  is a constant and  $P_1(x),P_2(x)\cdots, P_{g(n)}(x)$  are non-constant polynomials with integer coefficients and  $g(n)$  is as large as possible. For all  $2< n< 101$ , let  $t$  be the minimum possible value of  $g(n)$  in the above expression; for how many  $k$  in the specified range is  $g(k)=t$ ?

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