
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 a set $A=\left\{A_{1}, A_{2}, \cdots, A_{i}\right\}$, every pair of subsets $A_{i}, A_{j}$ satisfies
$$
\left|A_{i} \triangle A_{j}\right|=k.
$$

When $k=\frac{n+1}{2}$, $t \leq n+1$. For other values of $k$, $t \leq n$.

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