
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
The table below shows the quality inspection results of a batch of $N95$ masks produced by a factory's quality inspection department.

| Number of Masks Tested | $500$ | $1000$ | $1500$ | $2000$ | $3000$ | $4000$ |
|-------------------------|-------|--------|--------|--------|--------|--------|
| Number of Qualified Masks | $471$ | $946$ | $1425$ | $1898$ | $2853$ | $3812$ |
| Qualified Rate | $0.942$ | $0.946$ | $0.950$ | $a$ | $b$ | $0.953$ |

$(1)$ Find the values of $a=$______ and $b=$______;  
$(2)$ The estimated probability of randomly selecting a qualified mask from this batch of masks is ______ (rounded to $0.01$);  
$(3)$ If the factory needs to produce $285000$ qualified $N95$ masks, how many $N95$ masks does the factory estimate it needs to produce?

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