
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 celebration event at a certain school, a "ring toss game" was designed with the following rules: each person has $3$ rings to toss towards two targets $M$ and $N$. The player first tosses towards target $M$ once, scoring $1$ point for a successful ring toss and no points for a miss. Then, the player tosses towards target $N$ twice in a row, scoring $2$ points for each successful ring toss and no points for a miss. Prizes are awarded based on the cumulative score. It is known that for each toss, the probability of hitting target $M$ is $\frac{3}{4}$ and the probability of hitting target $N$ is $\frac{2}{3}$. Assuming the results of each toss by a player are independent, let $X$ denote the cumulative score.<br/>$(1)$ Find the probability that a player hits exactly $2$ times;<br/>$(2)$ Find the distribution table of $X$ and its expected value.

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