
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
What is the remainder in the following division:

$$
\left(x^{1001}-1\right):\left(x^{4}+x^{3}+2 x^{2}+x+1\right)
$$

and what if the divisor is the following polynomial:

$$
x^{8}+x^{6}+2 x^{4}+x^{2}+1
$$

(understanding remainder as the polynomial that arises after determining the last term of the quotient with a non-negative exponent).

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