
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 fair 10-sided die is rolled. If I roll $n$ and $n \leq 5$, I can choose to roll once more. Each roll result is independent. I win $\frac{10-n}{2}$ dollars if I roll once, and if I choose to roll a second die, I win the average of $(10-n)$ and $(10-m)$ dollars, where $m$ is the result of the second roll. What is the expected value of my win, in dollars?

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