
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 is a wooden  $3 \times 3 \times 3$  cube and 18 rectangular  $3 \times 1$  paper strips.  Each strip has two dotted lines dividing it into three unit squares.  The full surface of the cube is covered with the given strips, flat or bent.  Each flat strip is on one face of the cube.  Each bent strip (bent at one of its dotted lines) is on two adjacent faces of the cube.  What is the greatest possible number of bent strips?  Justify your answer.

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