
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 reals for which
\begin{align*}
  (a+b)(a+c) &= bc + 2 
  (b+c)(b+a) &= ca + 5 
  (c+a)(c+b) &= ab + 9
\end{align*}
If  $abc = \frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m+n$ .

*Proposed by Evan Chen*

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