
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 domino is a  $1\times2$  or  $2\times 1$  rectangle. Diego wants to completely cover a  $6\times 6$  board using  $18$  dominoes. Determine the smallest positive integer  $k$  for which Diego can place  $k$  dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remaining dominoes.

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