
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 seagrass is a gramineous herbaceous plant with well-developed roots and a very high reproductive rate, which poses a great threat to the nearshore ecology. In order to eliminate the harm of seagrass as soon as possible, on October 24, 2022, the municipal government issued the "Implementation Plan for the Elimination of Seagrass in Putian City," making specific arrangements for the city's elimination campaign. A research team wanted to understand the depth of the root distribution of seagrass in two towns, A and B, and used a proportional stratified sampling method to extract samples. It is known that the sample size of town A is $m=12$, the sample mean is $\overline{x}=18$, and the sample variance is ${S}_{1}^{2}=19$; the sample size of town B is $n=18$, the sample mean is $y=36$, and the sample variance is ${S}_{2}^{2}=70$.  
$(1)$ Calculate the mean $\overline{z}$ and variance $S^{2}$ of the total sample composed of samples from the two towns;  
$(2)$ To create a strong atmosphere of "broad mobilization and full participation," towns A and B decided to hold a "Seagrass Elimination Training" competition, with one representative team from each town participating. The first match was held in town A as determined by drawing lots. The rules of the competition are:  
Each match continues until a winner is determined. The winning team gets $1$ point, and the losing team gets $0$ points. The next match is held in the town of the losing team. The team that reaches $2$ points first wins the competition.  
When the match is held in town A, the probability of the town A team winning is $\frac{3}{5}$, and when the match is held in town B, the probability of the town A team winning is $\frac{1}{2}$. Assuming that the results of each match are independent. Let the final score of the town A team be $X$, find $E\left(X\right)$.  
Reference data: $12\times 18^{2}=3888$, $18\times 36^{2}=23328$, $28.8^{2}=829.44$, $12\times 10.8^{2}=1399.68$, $18\times 7.2^{2}=933.12$.

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