
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
Find at least one set \( M \), consisting of 7 consecutive natural numbers, for which there exists a polynomial \( P(x) \) of the fifth degree with the following properties:

a) All coefficients of the polynomial \( P(x) \) are integers;

b) For five numbers \( k \in M \), including the smallest and the largest numbers, the equation \( P(k) = k \) holds;

c) \( P(k) = 0 \) for one number \( k \in M \).

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