
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
Denote  $U$  as the set of  $20$  diagonals of the regular polygon  $P_1P_2P_3P_4P_5P_6P_7P_8$ . 
Find the number of sets  $S$  which satisfies the following conditions.

1.  $S$  is a subset of  $U$ .

2. If  $P_iP_j \in S$  and  $P_j P_k \in S$ , and  $i \neq k$ ,  $P_iP_k \in S$ .

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