
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 lottery bag, there are four types of balls: red, yellow, white, and black. Except for the color, all the balls are the same. Drawing one ball from the bag, red, yellow, and white represent first, second, and third prizes respectively, while black represents a consolation prize.  
$(1)$ If Xiaoming has one chance to participate in the lottery, Xiaoming winning is a ______ event (fill in "random", "certain", "impossible");  
$(2)$ If there are a total of $24$ balls in the bag, including $3$ red balls, $6$ yellow balls, and $9$ black balls, then in one draw,  
the probability of drawing the first prize is ______;  
the probability of drawing the second prize is ______;  
the probability of winning is ______;  
$(3)$ Now, with enough balls, please select $15$ balls to design a ball-drawing game, so that the probability of drawing a red ball is equal to the probability of drawing a white ball, and both are greater than the probability of drawing a black ball.

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