
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
If \( k < r \) and \( \mathscr{A} = \{ A_1, A_2, \cdots, A_t \} \) is a family of \( H_k \) such that each \( A_i \) is an \( r \)-element set \( (1 \leq i \leq t) \), then
$$
t \leq C_{n-1}^{r-1}.
$$

The equality holds if and only if \( \mathscr{A} \) is composed of all \( r \)-element subsets containing a certain element \( x \in X \).

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