
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 $2 \leq r \leq \frac{n}{2}$, and let $\mathscr{A}$ be a family of $r$-element subsets of $Z = \{1, 2, \ldots, n\}$. If $\mathscr{A} = \{A \mid A$ is an $r$-element subset of $Z$ that contains a fixed element $x \in Z\}$, then equality holds.

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