
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 version of the game, Allie and Betty still take turns rolling a standard die. The score for each roll is now determined by \(f(n)\), where:
\[f(n) = \left\{
\begin{array}{cl}
8 & \text{ if } n \text{ is a perfect square}, \\
4 & \text{ if } n \text{ is a multiple of 3 but not a perfect square}, \\
1 & \text{ if } n \text{ is a multiple of 2 but not a perfect square}, \\
0 & \text{ otherwise}.
\end{array}
\right.\]
Allie rolls the die four times and gets a 3, 4, 6, and 1. Betty rolls and gets 4, 2, 5, and 1. What is the product of Allie's total points and Betty's total points?

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