
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
A polynomial  $p(x)$  of degree  $n\ge 2$  has exactly  $n$  real roots, counted with multiplicity. We know that the coefficient of  $x^n$  is  $1$ , all the roots are less than or equal to  $1$ , and  $p(2)=3^n$ . What values can  $p(1)$  take?

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