
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
For every natural number  $n\geq{2}$  consider the following affirmation  $P_n$ :
"Consider a polynomial  $P(X)$  (of degree  $n$ ) with real coefficients. If its derivative  $P'(X)$  has  $n-1$  distinct real roots, then there is a real number  $C$  such that the equation  $P(x)=C$  has  $n$  real,distinct roots." 
Are  $P_4$  and  $P_5$  both true? Justify your answer.

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