
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 a total of $4$ balls of different colors, black and white. A study group conducted a ball-picking experiment by randomly picking a ball from the bag after stirring it, noting the color, and then putting it back in the bag. The following table shows a set of statistical data during the activity:<br/>

| Number of picks $n$ | $2048$ | $4040$ | $10000$ | $12000$ | $24000$ |
|----------------------|--------|--------|---------|---------|---------|
| Number of white balls picked $m$ | $1061$ | $2048$ | $4979$ | $6019$ | $12012$ |
| Frequency of picking a white ball $\frac{m}{n}$ | $0.518$ | $0.5069$ | $0.4979$ | $0.5016$ | $0.5005$ |

$(1)$ Please estimate: when $n$ is very large, the frequency of picking a white ball will be close to ______ (rounded to $0.1$);<br/>
$(2)$ Estimate how many white balls are in the bag?<br/>
$(3)$ If one ball is picked first without replacement, and then another ball is picked, please use a list or tree diagram method (choose only one) to find the probability of picking two balls of the same color.

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