
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
For each positive integer  $ k$ , find the smallest number  $ n_{k}$  for which there exist real  $ n_{k}\times n_{k}$  matrices  $ A_{1}, A_{2}, \ldots, A_{k}$  such that all of the following conditions hold:

(1)  $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$ ,

(2)  $ A_{i}A_{j}= A_{j}A_{i}$  for all  $ 1 \le i, j \le k$ , and 

(3)  $ A_{1}A_{2}\ldots A_{k}\ne 0$ .

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