
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
There are twelve tiles, numbered from $1$ to $12$. Four players each randomly select and keep three of the tiles and sum those values. What is the probability that all four players obtain an odd sum? Express the probability in the simplest fractional form $m/n$ where $m$ and $n$ are relatively prime positive integers and find $m+n$.

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