
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
Anastasia is taking a walk in the plane, starting from \((1,0)\). Each second, if she is at \((x, y)\), she moves to one of the points \((x-1, y)\), \((x+1, y)\), \((x, y-1)\), and \((x, y+1)\), each with \(\frac{1}{4}\) probability. She stops as soon as she hits a point of the form \((k, k)\). What is the probability that \(k\) is divisible by 3 when she stops?

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