
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
A unique four-sided die is designed to show numbers 1, 2, 3, and 4 with probabilities $p_1 = \frac{1}{4}$, $p_2 = \frac{1}{4}$, $p_3 = \frac{1}{6}$, $p_4 = \frac{1}{3}$ respectively. 
- If the die shows a 1, you win 2 dollars.
- If the die shows a 2 and previously has shown an odd number, you lose 3 dollars; otherwise, you lose nothing.
- If it shows a 3, you win nothing.
- If it shows a 4, you win 5 dollars.
Assuming the die is thrown twice, what is the expected return from playing this game? Each throw is independent of the previous.

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