
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
Consider an urn that contains one black ball, four white balls, and some blue balls. A ball is randomly drawn from the urn, its color is observed, and the ball is returned to the urn. Then another ball is randomly drawn from the urn. For which quantities of blue balls is the probability that the two drawn balls have the same color equal to $1 / 2$?

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