
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 box, there are a total of $40$ identical black and white balls except for color. Xiao Ming conducts a ball-picking experiment. After stirring the balls in the box, he randomly picks out a ball, notes its color, and then puts it back into the box. This process is repeated continuously. The following table shows a set of statistical data from the experiment:

| Number of picks $n$ | $100$ | $200$ | $300$ | $500$ | $800$ | $1000$ | $3000$ |
|---------------------|-------|-------|-------|-------|-------|--------|--------|
| Number of white balls picked $m$ | $70$ | $128$ | $171$ | $302$ | $481$ | $599$ | $903$ |
| Frequency of picking white balls $\frac{m}{n}$ | $0.75$ | $0.64$ | $0.57$ | $0.604$ | $0.601$ | $0.599$ | $0.602$ |

$(1)$ Estimate: When $n$ is very large, the probability of picking a white ball is approximately ______. (Round to $0.1$)<br/>
$(2)$ Estimate the number of white balls in the box to be ______.<br/>
$(3)$ If $x$ additional balls, identical to the black and white balls except for color, are added to the box, and out of these $x$ balls only $1$ is white, after stirring the balls uniformly and picking one ball at random each time, it is found through a large number of repeated ball-picking experiments that the frequency of picking a white ball stabilizes at $50\%$. In this case, the most likely value for $x$ is ______.

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