
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 at the center dot labeled $C$ on a 3x3 grid where the following special condition holds: Edges exist between directly neighboring dots (vertically and horizontally), plus each corner dot is connected directly to the center dot (meaning it can move directly from any corner to $C$ or vice versa). The ant moves randomly every minute. What is the probability that after 6 minutes, the ant returns to center $C$? Label the center of grid $C$ and upper middle grid point as $D$. Assume a symmetric distribution of movements.

```
         D
   * - * - *
   | X | X |
   * - C - *
   | X | X |
   * - * - *
```

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