
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 rules of the ball-drawing experiment are as follows: There is initially one white ball and one red ball in the box. In each round, one ball is drawn with replacement. If two white balls are drawn consecutively in a round, the round is considered successful; otherwise, it is considered a failure. During the drawing process, if a round is successful, the drawing stops; otherwise, a red ball is added back to the box, and the next round of drawing continues.  
$(1)$ When a person conducts the ball-drawing experiment, they can have a maximum of three rounds. Even if the third round is not successful, the drawing stops. Let the number of rounds the person conducts be a random variable $X$. Find the probability distribution and mathematical expectation of $X$.  
$(2)$ To verify that the probability of a successful ball-drawing experiment does not exceed $\frac{1}{2}$, 1000 math enthusiasts independently conduct the experiment. Let $t$ represent the number of rounds when success is achieved, and $y$ represent the corresponding number of people. Some statistical data is given as follows:  

| $t$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|-----|-----|-----|-----|-----|-----|
| $y$ | $232$ | $98$ | $60$ | $40$ | $20$ |

Find the regression equation of $y$ with respect to $t$, $\hat{y}=\frac{\hat{b}}{t}+\hat{a}$, and predict the total number of successful people (rounded to the nearest whole number).  
Given: Empirical regression equation coefficients: $\hat{b}=\frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}-n\overline{x}\cdot\overline{y}}{\sum_{i=1}^{n}{x}_{i}^{2}-n{\overline{x}}^{2}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$; Reference data: $\sum_{i=1}^{5}{x}_{i}^{2}=1.46$, $\overline{x}=0.46$, ${\overline{x}}^{2}=0.212$ (where ${x}_{i}=\frac{1}{{t}_{i}}$, $\overline{x}=\frac{1}{5}\sum_{i=1}^{5}{x}_{i}$).

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