
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
George now has a modified six-sided die. The probability that it rolls a 6 is $\frac{3}{8}$, the probability that it rolls a 4 is $\frac{1}{4}$, and the probability that it rolls any other number (1, 2, 3, 5) is $\frac{1}{20}$ each. The die is rolled repeatedly until a 6 is rolled for the first time. Calculate the expected value of the sum of numbers shown until a 6 appears for the first time.

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