
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
On the board below, it is permitted to move any object from its square to any adjacent empty square above, below, sideways, or diagonally.

a) Show how to swap the position of all the hats with all the trophies in just five moves. Argue why it is not possible to swap them with fewer than five moves.

b) On the other board shown below, what is the minimum number of moves to swap the hats' positions with the trophies?

c) If it were a board similar to the previous ones, but with 1000 hats and 1000 trophies, what would be the minimum number of moves to swap them?


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