
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 dice rolls numbers from 1 to 6, but it has a higher chance of rolling odd numbers. The probabilities are as follows: rolling any odd number (1, 3, or 5) has a probability of $\frac{1}{3}$ each, while even numbers (2, 4, or 6) have probabilities of $\frac{1}{9}$, $\frac{1}{18}$, and $\frac{1}{9}$ respectively. Rolling an odd number gains $\$4$ and rolling an even number loses $\$6$. Determine the expected monetary outcome of a dice roll.

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