
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 biased die shows a 6 with a probability of $\frac{1}{2}$, any number from 1 to 5 with equal probability. Rolling a 6 gains $\$5$, and rolling any number from 1 to 4 gains $\$1$, but rolling a 5 loses $\$10$. What is the expected value of rolling this die?

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