
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
One face of a tetrahedron has sides of length  $3$ ,  $4$ , and  $5$ . The tetrahedron’s volume is  $24$  and surface area is  $n$ . When  $n$  is minimized, it can be expressed in the form  $n = a\sqrt{b} + c$ , where  $a$ ,  $b$ , and  $c$  are positive integers and b is not divisible by the square of any prime. Evaluate  $a + b + c$ .

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