
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 coin comes up heads with probability $\frac{1}{3}$, tails with probability $\frac{1}{2}$, and rests on its edge with probability $\frac{1}{6}$. If it comes up heads, you win 2 dollars. If it comes up tails, you win 4 dollars. If it lands on its edge, you lose 6 dollars. What is the expected winnings from flipping this coin? Express your answer as a dollar value, rounded to the nearest cent.

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