
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  $S$  be the set of triples  $(a,b,c)$  of non-negative integers with  $a+b+c$  even. The value of the sum
    \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
    can be expressed as  $\frac{m}{n}$  for relative prime positive integers  $m$  and  $n$ . Compute  $m+n$ .

*Proposed by Nathan Xiong*

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