
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
We know that \( P(x) \) is a polynomial of degree \( n \), and that the function 

\[ 
f(x) = P(x) \cdot \sum_{i=1}^{n} |x-i| 
\]

is differentiable for all real \( x \). Determine all polynomials with this property.

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