
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
In an opaque bag, there is $1$ yellow ball (denoted as $Y$) and $2$ white balls (denoted as $W_{1}$ and $W_{2}$ respectively). These balls are identical except for their colors. They are thoroughly mixed.<br/>$(1)$ The probability of drawing exactly one white ball from the bag is.<br/>$(2)$ A ball is drawn from the bag, replaced, mixed again, and then another ball is drawn. Using a tree diagram or a list, find the probability that the colors of the two balls drawn are the same.

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