
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
An ant starts out at \((0,0)\). Each second, if it is currently at the square \((x, y)\), it can move to \((x-1, y-1)\), \((x-1, y+1)\), \((x+1, y-1)\), or \((x+1, y+1)\). In how many ways can it end up at \((2010,2010)\) after 4020 seconds?

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