
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 strange die, when rolled, shows a number 1 with probability \(\frac{1}{4}\), a number 2 with probability \(\frac{1}{4}\), a number 3 with probability \(\frac{1}{6}\), a number 4 with probability \(\frac{1}{6}\), a number 5 with probability \(\frac{1}{6}\), and a number 6 with probability \(\frac{1}{12}\). If it shows 1, you gain \$2. If it shows 2, you gain \$2. If it shows 3, you gain \$4. If it shows 4, you gain \$4. If it shows 5, you lose \$6. If it shows 6, you lose \$12. Compute the expected winnings from rolling this die.

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