
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
The roots of a monic cubic polynomial  $p$  are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to  $10$ . Under these conditions, the largest possible value of  $|p(-1)|$  can be written as  $\frac{m}{n}$ , where  $m$ ,  $n$  are relatively prime integers. Find  $m + n$ .

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