
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 are $3$ black balls and $2$ white balls, all of which are identical except for color.  
$(1)$ Using a tree diagram or list method, find the probability of randomly drawing two black balls from the bag.  
$(2)$ If $x$ white balls and $y$ black balls are added to the bag, and the probability of drawing a white ball is $\frac{1}{3}$, determine the relationship between $x$ and $y$.

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