
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 robot is located on an infinite grid and can be given the following commands:

- up (the robot moves to the cell above),
- down (the robot moves to the cell below),
- left (the robot moves to the cell to the left),
- right (the robot moves to the cell to the right).

If, for example, the robot executes a sequence of four commands (up, right, down, left), it is obvious that it will return to its initial position, i.e., it will be in the same cell from which it started moving. How many different sequences of 8 commands will return the robot to its initial position?

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