
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 $19$th Asian Games were held in Hangzhou from September 23 to October 8, 2023. In order to promote the Olympic and Asian Games spirit and enhance physical exercise awareness, a school organized a badminton competition. It is known that in the badminton singles rules: if the serving side wins, the serving side gets $1$ point and continues to serve in the next round; if the receiving side wins, the receiving side gets $1$ point and becomes the serving side in the next round. Now, two players, A and B, are playing badminton singles. If A serves, the probability of A scoring is $\frac{3}{5}$ and the probability of B scoring is $\frac{2}{5}$; if B serves, the probability of B scoring is $\frac{4}{5}$ and the probability of A scoring is $\frac{1}{5}$. The results of each round are independent of each other. It was decided by drawing lots that A would serve in the first round.  
$(1)$ Find the probability of A serving in the third round;  
$(2)$ Let $X$ be the total score of A in the first three rounds, find the distribution and expectation of $X$.

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