
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 all sequences  $a_{0}, a_{1},\ldots, a_{n}$  of real numbers such that  $a_{n}\neq 0$ , for which the following statement is true:

  If  $f: \mathbb{R}\to\mathbb{R}$  is an  $n$  times differentiable function
  and  $x_{0}<x_{1}<\ldots <x_{n}$  are real numbers such that
    $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$  then there is  $h\in (x_{0}, x_{n})$  for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]

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