
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 from $1$ to $12$, respectively. Each of four players randomly selects and keeps three of the tiles, and sums those three values. Evaluate the probability that all four players obtain an odd sum. Express your final answer as a sum of relatively prime positive integers $m+n$.

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