
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 prime number and  $F=\left \{0,1,2,...,p-1 \right \}$ . Let  $A$  be a proper subset of  $F$  that satisfies the following property:  if  $a,b \in A$ , then  $ab+1$  (mod  $p$ )  $ \in A$ .
How many elements can  $A$  have? (Justify your answer.)


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