
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 peculiar die, when rolled, shows a 6 with probability $\frac{1}{4}$, any number from 1 to 5 with probability $\frac{1}{5}$ each. If it shows a 6, you win 4 dollars. If it shows an even number (2 or 4), you lose 2 dollars. If it shows an odd number (1, 3, or 5), you gain 1 dollar. What is your expected winnings from rolling this die?

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