
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 square with a side length of 4 cm is divided by three parallel horizontal and three parallel vertical lines into 16 smaller squares with a side length of 1 cm. The sides of these smaller squares, including those on the boundary of the large square, will be called unit segments. In how many ways can an orientation be assigned to each of the 40 unit segments such that the total sum of all 40 vectors is zero? The answer can be given in the form of a formula; it is not necessary to compute the numerical value.

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