
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
Dirock has a very neat rectangular backyard that can be represented as a  $32\times 32$  grid of unit squares. The rows and columns are each numbered  $1,2,\ldots, 32$ . Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by  $3$ . Dirock would like to build a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of the yard such that
 
- The fence does not pass through any grid squares containing rocks;
- The interior of the fence contains exactly 5 rocks.

In how many ways can this be done?

*Ray Li*

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