
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]


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A certain school is preparing to establish a football club. Due to the large number of applicants, a "penalty kick test" is required to determine whether to admit them. The rules are as follows: each person has up to four chances, and as long as they score $2$ penalty kicks in a row, they will stop playing and be admitted; if it is already determined that they cannot score $2$ penalty kicks in a row, they will stop playing and not be admitted. The table below shows the training data of a student for six sessions, with the goal frequency in these $150$ penalty kicks representing the probability of scoring in a single penalty kick.

| Number of Penalty Kicks | $20$ | $30$ | $30$ | $25$ | $20$ | $25$ |
|-------------------------|------|------|------|------|------|------|
| Number of Goals         | $15$ | $17$ | $22$ | $18$ | $14$ | $14$ |

$(1)$ Find the probability of this student being admitted;<br/>$(2)$ If this student is to take the "penalty kick test," let the number of goals he scores in the test be $X$, find the expected value of the random variable $X$.

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