
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
At a temple fair, there is a game called "ring toss". The rules are as follows: each group has 3 rings, and they throw them towards two targets, $A$ and $B$. They first throw two rings consecutively towards target $A$, scoring 1 point for each successful toss and 0 points for misses. Then, they throw one ring towards target $B$, scoring 2 points for each successful toss and 0 points for misses. The prizes are awarded based on the final score obtained. It is known that the probability of hitting target $A$ for player $A$ is $\frac{3}{4}$ for each throw, and the probability of hitting target $B$ is $\frac{1}{2}$. It is assumed that the results of each throw by player $A$ are independent of each other.  
$(1)$ Find the probability that player $A$ hits the target exactly once in one round of the game.  
$(2)$ Find the distribution table and the expected value of the total score $X$ for player $A$ in one round of the game.  
$(3)$ Player $A$ played 5 rounds of the game in a row. Assuming that the results of each round are independent, find the probability that player $A$ scored 2 or 3 points in exactly 3 rounds of the game.

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