
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
Call a polynomial  $P$  *prime-covering* if for every prime  $p$ , there exists an integer  $n$  for which  $p$  divides  $P(n)$ . Determine the number of ordered triples of integers  $(a,b,c)$ , with  $1\leq a < b < c \leq 25$ , for which  $P(x)=(x^2-a)(x^2-b)(x^2-c)$  is prime-covering.

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