
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
Twelve tiles are numbered $1, 2, 3, \cdots, 12$. Half of these tiles are odd numbered and half are even numbered. Each of three players randomly selects and keeps four of the tiles and sums those four values. Find the probability that all three players obtain an odd sum using the format $m/n$, where $m$ and $n$ are relatively prime integers.

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