
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  $A$  be a given set with  $n$  elements. Let  $k<n$  be a given positive integer. Find the maximum value of  $m$  for which it is possible to choose sets  $B_i$  and  $C_i$  for  $i=1,2,\ldots,m$  satisfying the following conditions:
[list=1]
[*] $B_i\subset A,$   $|B_i|=k,$ [*] $C_i\subset B_i$  (there is no additional condition for the number of elements in  $C_i$ ), and
[*] $B_i\cap C_j\neq B_j\cap C_i$  for all  $i\neq j.$ [/list]

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