
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  $ n$  and  $ k$  be given natural numbers, and let  $ A$  be a set such that \[ |A| \leq \frac{n(n+1)}{k+1}.\] For  $ i=1,2,...,n+1$ , let  $ A_i$  be sets of size  $ n$  such that \[ |A_i \cap A_j| \leq k \;(i \not=j)\ ,\] \[ A=  \bigcup_{i=1}^{n+1} A_i.\] Determine the cardinality of  $ A$ . 

*K. Corradi*

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