
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
A certain shopping mall has a game with the following rules: In an opaque box containing $8$ red balls and some white balls (each ball is identical except for color), a random ball is drawn. If a red ball is drawn, the player wins a bottle of drink. The staff counted the number of people who participated in the game and the number of people who won a drink (see table below).

| Number of Participants | $200$ | $300$ | $400$ | $500$ |
|------------------------|-------|-------|-------|-------|
| Number of Winners      | $39$  | $63$  | $82$  | $99$  |
| Winning Frequency      |       |       |       |       |

$(1)$ Calculate and complete the table;  
$(2)$ Estimate the probability of winning a drink as ______;  
$(3)$ Please estimate the number of white balls in the bag.

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