
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
Given a finite nonempty set  $X$  with real values, let  $f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a$ , where  $\left\lvert X \right\rvert$  denotes the cardinality of  $X$ .  For ordered pairs of sets  $(A,B)$  such that  $A\cup B = \{1, 2, \dots , 100\}$  and  $A\cap B = \emptyset$  where  $1\leq |A| \leq 98$ , select some  $p\in B$ , and let  $A_{p} = A\cup \{p\}$  and  $B_{p} = B - \{p\}.$  Over all such  $(A,B)$  and  $p\in B$  determine the maximum possible value of  $(f(A_{p})-f(A))(f(B_{p})-f(B)).$ 

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