
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
Let  $P$  be a set of  $7$  different prime numbers and  $C$  a set of  $28$  different composite numbers each of which is a product of two (not necessarily different) numbers from  $P$ . The set  $C$  is divided into  $7$  disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of  $C$  are there ?

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