
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
The plane is divided by lines into square cells with side length $a$. The cells are numbered using pairs of integers $(m, n)$ (where the cell immediately to the right has the number $(m+1, n)$, and the cell immediately above has the number $(m, n+1)$). Find the probability that a coin of radius $r < a/2$ randomly tossed onto the plane will completely fall into a cell for which $m+n$ is divisible by three.

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