
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 a pocket, there are $4$ red balls and $8$ white balls, which are identical except for color.  
$(1)$ Determine the event "drawing a black ball randomly from the pocket" and write down the probability of its occurrence.  
$(2)$ Find the probability of drawing a red ball randomly from the pocket.  
$(3)$ Now, some white balls are removed from the pocket and the same number of red balls are added. After thorough mixing, the probability of drawing a red ball from the pocket is $\frac{5}{6}$. How many white balls were removed?

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