
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 the set of numbers $P=\left\{a_{1}, a_{2}, \cdots, a_{m}\right\}$, with its mean $C_{P}=\frac{a_{1}+a_{2}+\cdots+a_{m}}{m}$. Now, divide the set $S=\{1,2, \cdots, n\}$ into two non-empty disjoint subsets $A$ and $B$. Find the maximum value of $\left|C_{A}-C_{B}\right|$, and discuss the number of different ordered pairs $(A, B)$ when the maximum value is achieved.

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