
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 interger  $n\geq 4$ , we consider the  $m$  subsets  $A_1, A_2,\dots, A_m$  of  $\{1, 2, 3,\dots, n\}$ , such that $A_1$  has exactly one element,  $A_2$  has exactly two elements,....,  $A_m$  has exactly  $m$  elements and none of these subsets is contained in any other set. Find the maximum value of  $m$ .

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