
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 the set \( \mathrm{T} = \{1, 2, \ldots, 999\} \), find the maximum number \( \mathrm{k} \) of different subsets \( \mathrm{A}_1, \mathrm{A}_2, \ldots, \mathrm{A}_\mathrm{k} \) such that for any \( \mathrm{i}, \mathrm{j} \) with \( 1 \leq \mathrm{i} < \mathrm{j} \leq \mathrm{k} \), we have \( A_i \cup A_j = \mathrm{T} \).

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