
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 that \( A_{1}, A_{2}, \cdots, A_{t} \) and \( B_{1}, B_{2}, \cdots, B_{t} \) are subsets of an \( n \)-element set \( X \) satisfying:
(i) \( A_{i} \cap B_{i} = \varnothing \quad (i = 1, 2, \cdots, t) \);
(ii) If \( i \neq j \), \( A_{i} \) is not a subset of \( A_{j} \cup B_{j} \) ( \( i, j = 1, 2, \cdots, t \) ).

Let \( \left| A_{i} \right| = a_{i} \) and \( \left| B_{i} \right| = b_{i} \),
\[ w(i) = w \left( a_{i}, b_{i}, n \right) = \frac{1}{C_{n_{i} \cdot o_{i}}}, \]

then
\[ \sum_{i=1}^{t} w(i) \leqslant 1. \]

Equality holds if and only if \( B_{1} = B_{2} = \cdots = B_{t} = B \) and \( A_{1}, A_{2}, \cdots, A_{t} \) are all the \( a \)-element subsets of \( X - B \) \( (1 \leqslant a \leqslant n - |B|) \).

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