
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  $a$ ,  $b$ ,  $c$  be positive real numbers for which \[
  \frac{5}{a} = b+c, \quad
  \frac{10}{b} = c+a, \quad \text{and} \quad
  \frac{13}{c} = a+b. \] If  $a+b+c = \frac mn$  for relatively prime positive integers  $m$  and  $n$ , compute  $m+n$ .

*Proposed by Evan Chen*

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