
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
Suppose  $S_n$  is the set of positive divisors of  $n$ , and denote  $|X|$  as the number of elements in a set  $X$ . Let  $\xi$  be the set of positive integers  $n$  where  $|S_n| = 2m$  is even, and  $S_n$  can be partitioned evenly into pairs  $\{a_i, b_i\}$  for integers  $1 \le i \le m$  such that the following conditions hold: $\bullet$   $a_i$  and  $b_i$  are relatively prime for all integers  $1 \le i  \le m$  $\bullet$  There exists a  $j$  where  $6$  divides  $a^2_j+ b^2_j+ 1$  $\bullet$   $|S_n| \ge 20.$ Determine the number of positive divisors  $d | 24!$  such that  $d \in \xi$ .

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