
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 Mediterranean polynomial has only real roots and it is of the form
\[ P(x) =  x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients  $a_0\ldots,a_7$ .  Determine the largest real number that occurs as a root of some Mediterranean polynomial.
*(Proposed by Gerhard Woeginger, Austria)*

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