
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 $n$ identical red balls. In order to estimate the number of red balls in the bag, eight students conducted a ball-touching experiment using mathematical grouping: $10$ white balls of the same size and shape as the red balls were placed in the bag. After mixing them up, one ball was randomly picked out, its color noted, and then placed back in the bag. This process was repeated multiple times, and the class data summary table was obtained as shown below:

| Number of Picks | $150$ | $300$ | $600$ | $900$ | $1200$ | $1500$ |
|-----------------|-------|-------|-------|-------|--------|--------|
| Frequency of Picking White Ball | $60$ | $a$ | $247$ | $365$ | $484$ | $609$ |
| Frequency of Picking White Ball | $0.400$ | $0.42$ | $0.412$ | $0.406$ | $0.403$ | $b$ |

$(1)$ According to the table data format, $a=$______, $b=$______;<br/>$(2)$ Please calculate: the probability of picking a red ball is ______ (rounded to $0.1$);<br/>$(3)$ Estimate: the value of $n$, the number of red balls in the opaque bag.

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