
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  $A$  and  $B$  be two non-empty subsets of  $X = \{1, 2, . . . , 8 \}$  with  $A \cup B = X$  and  $A \cap B = \emptyset$ . Let  $P_A$  be the product of all elements of  $A$  and let  $P_B$  be the product of all elements of  $B$ . Find the minimum possible value of sum  $P_A +P_B$ . 

PS. It is a variation of [JBMO Shortlist 2019 A3 ](https://artofproblemsolving.com/community/c6h2267998p17621980)

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