
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 triple integer  $(a, b, c)$  is called *brilliant* when it satisfies:
(i)  $a> b> c$  are prime numbers
(ii)  $a = b + 2c$ (iii)  $a + b + c$  is a perfect square number
Find the minimum value of  $abc$  if triple   $(a, b, c)$  is *brilliant*.

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