
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
A right rectangular prism has integer side lengths  $a$ ,  $b$ , and  $c$ . If  $\text{lcm}(a,b)=72$ ,  $\text{lcm}(a,c)=24$ , and  $\text{lcm}(b,c)=18$ , what is the sum of the minimum and maximum possible volumes of the prism?

*Proposed by Deyuan Li and Andrew Milas*

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