
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 weighted dice has six faces. The probability of rolling a 2 is $\frac{1}{4}$, the probability of rolling a 5 is $\frac{1}{2}$, and the probability of rolling any other number is $\frac{1}{12}$ for each number. You win $\$4$ if you roll a 2, $\$6$ if you roll a 5, and lose $\$3$ for rolling any other number. What is the expected value, in dollars, of rolling this dice once?

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