
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 \( P \in \mathbb{K}[X] \) and \((a, b) \in \mathbb{K}^{2}\) such that \( a \neq b \). Given that the remainder of the Euclidean division of \( P \) by \( X-a \) is 1 and the remainder of the Euclidean division of \( P \) by \( X-b \) is -1, determine the remainder of the Euclidean division of \( P \) by \((X-a)(X-b)\).

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