
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 weighted coin has the following probabilities: coming up heads with a probability of $\frac{3}{7}$, tails with a probability of $\frac{1}{4}$, resting on its edge with a probability of $\frac{1}{7}$, and landing on another side with a probability of $\frac{2}{7}$. The winnings are as follows: win 2 dollars for heads, 4 dollars for tails, lose 6 dollars if it lands on its edge, and lose 2 dollars if it lands on the other side. What is the expected winnings from flipping this coin?

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