
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
1. Find the quotient and the remainder of the Euclidean division of \( X^{4} + 3X^{2} + 2X + 1 \) by \( X^{2} - 2 \).
2. Let \( n \in \mathbb{N}^{*} \). Find the remainder of the Euclidean division of \( X^{n} - X + 1 \) by \( X^{2} - 1 \).

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