
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
In a modified game, Allie and Betty again take turns rolling a standard die but now the score function has changed to \[f(n) = \left\{
\begin{array}{cl} 
9 & \text{if } n \text{ is a multiple of 3}, \\
3 & \text{if } n \text{ is only a multiple of 2}, \\
1 & \text{if } n \text{ is odd but not a multiple of 3}, \\
0 & \text{otherwise}.
\end{array}
\right.\]
Allie rolls the die five times and gets 5, 2, 6, 1, and 3. Betty rolls and gets 6, 4, 1, 2, and 5. Calculate the product of Allie's and Betty's total points.

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