
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 ten 5-element sets \( A_{1}, A_{2}, \cdots, A_{10} \), where the intersection of any two sets contains at least two elements. Let \( A=\bigcup_{i=1}^{10} A_{i}=\left\{x_{1}, x_{2}, \cdots, x_{n}\right\} \). For any \( x_{i} \in A \), let \( k_{i}(i=1,2, \cdots, n) \) be the number of sets among \( A_{1}, A_{2}, \cdots, A_{10} \) that contain the element \( x_{i} \). Define \( m = \max \left\{k_{1}, k_{2}, \cdots, k_{n}\right\} \). Determine the minimum value of \( m \).

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