
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  $a$ ,  $b$ , and  $c$  be non-zero real number such that  $\tfrac{ab}{a+b}=3$ ,  $\tfrac{bc}{b+c}=4$ , and  $\tfrac{ca}{c+a}=5$ . There are relatively prime positive integers  $m$  and  $n$  so that  $\tfrac{abc}{ab+bc+ca}=\tfrac{m}{n}$ . Find  $m+n$ .

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