
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 coin with unusual probabilities comes up heads with probability $\frac13$, tails with probability $\frac16$, and rests on its edge with probability $\frac12$. If it comes up heads, I win a random amount which can be either 2 dollars or 3 dollars with equal probability. If it comes up tails, I win 4 dollars. However, if it lands on its edge, I lose 7 dollars. What is the expected winnings from flipping this coin?

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