
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
Let  $P(t) = t^3+27t^2+199t+432$ .  Suppose  $a$ ,  $b$ ,  $c$ , and  $x$  are distinct positive reals such that  $P(-a)=P(-b)=P(-c)=0$ , and \[
\sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If  $x=\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , compute  $m+n$ .

*Proposed by Evan Chen*

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