
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 in one of the cells of an infinite grid paper and can be given the following commands:
- move up (the robot moves to the adjacent cell above);
- move down (the robot moves to the adjacent cell below);
- move left (the robot moves to the adjacent cell to the left);
- move right (the robot moves to the adjacent cell to the right).

For example, if the robot executes a sequence of four commands (up, right, down, left), it will obviously return to its initial position, which means it will be in the same cell from which it started moving. How many different sequences of 8 commands return the robot to its initial position?

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