
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
Find the value of 
$$
\sum_{1 \leq a<b<c} \frac{1}{2^{a} 3^{b} 5^{c}}
$$
(i.e. the sum of \(\frac{1}{2^{a} 3^{b} 5^{c}}\) over all triples of positive integers \((a, b, c)\) satisfying \(a < b < c\)).

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