
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 \( a_{1}, \ldots a_{n} \) and \( b_{1}, \ldots, b_{n} \) be elements of \( \mathbb{K} \) (with the \( a_{i} \) pairwise distinct). Find all polynomials \( P \in \mathbb{K}[X] \) such that \( P(a_{i}) = b_{i} \).

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