
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 $2$ white balls, $3$ black balls, and $5$ red balls, each ball being identical except for color.<br/>$(1)$ Drawing a ball from the bag at random, the event of drawing a red ball is ______; the event of drawing a yellow ball is ______ (Fill in "impossible" or "certain" or "random").<br/>$(2)$ Drawing a ball from the bag at random, the probability of drawing a black ball is ______;<br/>$(3)$ Now, some additional black balls are added to the bag and mixed evenly with the original $10$ balls, such that the probability of drawing a black ball from the bag is $\frac{3}{4}$. Find the number of additional black balls added to the bag.

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