
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  $ 3\times3\times3$  cube composed of  $ 27$  unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a  $ 3\times3\times1$  block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a  $ 45^\circ$  angle with the horizontal plane.

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