
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 special die is designed such that the chance of rolling a 1 or 2 is $\frac{1}{4}$ each, and the chance of rolling any number from 3 to 6 is $\frac{1}{8}$ each. If you roll a 1, you earn \$4, but if you roll a 2, you lose \$3. Rolling any other number results in neither a gain nor a loss. What is the expected value, in dollars, of your earnings after one roll?

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