
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
We have 21 pieces of type $\Gamma$ (each formed by three small squares). We are allowed to place them on an $8 \times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added while following this rule. What is the smallest $k$ such that there exists a maximal arrangement of $k$ pieces of type $\Gamma$?

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