
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
 $A$  and  $B$  plays a game, with  $A$  choosing a positive integer  $n \in \{1, 2, \dots, 1001\} = S$ .  $B$  must guess the value of  $n$  by choosing several subsets of  $S$ , then  $A$  will tell  $B$  how many subsets  $n$  is in.  $B$  will do this three times selecting  $k_1, k_2$  then  $k_3$  subsets of  $S$  each.

What is the least value of  $k_1 + k_2 + k_3$  such that  $B$  has a strategy to correctly guess the value of  $n$  no matter what  $A$  chooses?

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