
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 real polynomials  $ p(x)$  of degree  $ n \ge 2$  for which there exist real numbers  $ r_1 < r_2 <  ... < r_n$  such that

(i)  $ p(r_i) \equal{} 0, 1 \le i \le n$ , and

(ii)  $ p' \left( \frac {r_i \plus{} r_{i \plus{} 1}}{2} \right) \equal{} 0, 1 \le i \le n \minus{} 1$ .

**Follow-up:**  In terms of  $ n$ , what is the maximum value of  $ k$  for which  $ k$  consecutive real roots of a polynomial  $ p(x)$  of degree  $ n$  can have this property?  (By "consecutive" I mean we order the real roots of  $ p(x)$  and ignore the complex roots.)  In particular, is  $ k \equal{} n \minus{} 1$  possible for  $ n \ge 3$ ?

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