
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
There are  $10$  cups, each having  $10$  pebbles in them. Two players  $A$  and  $B$  play a game, repeating the following in order each move: $\bullet$   $B$  takes one pebble from each cup and redistributes them as  $A$  wishes.  $\bullet$  After  $B$  distributes the pebbles, he tells how many pebbles are in each cup to  $A$ . Then  $B$  destroys all the cups having no pebbles. $\bullet$   $B$  switches the places of two cups without telling  $A$ .

After finitely many moves,  $A$  can guarantee that  $n$  cups are destroyed. Find the maximum possible value of  $n$ .
(Note that  $A$  doesn't see the cups while playing.)

*Proposed by Emre Osman*

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