
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 several identical balls of the same color except for the color. A math study group conducted a ball-touching experiment, stirring the balls and randomly picking one out to record its color, then putting it back in the bag, and repeating the process. The table below shows a set of statistical data during the activity:

| Number of times of picking a ball $n$ | $100$ | $150$ | $200$ | $500$ | $800$ | $1000$ |
|---------------------------------------|-------|-------|-------|-------|-------|--------|
| Number of times a white ball is picked $m$ | $59$ | $96$ | $b$ | $295$ | $480$ | $601$ |
| Frequency of picking a white ball $\frac{m}{n}$ | $a$ | $0.64$ | $0.58$ | $0.59$ | $0.60$ | $0.601$ |

$(1)$ In the table above, $a=$______, $b=$______;  
$(2)$ The estimated value of the probability of picking a white ball is ______ (rounded to $0.1$);  
$(3)$ If there are $12$ white balls in the bag, how many balls of other colors are there in the bag?

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