
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 wholesale market, the volleyball supply from Factory A accounts for $40\%$, from Factory B accounts for $30\%, and from Factory C accounts for $30\%$. The qualification rates of the volleyballs produced by Factory A, Factory B, and Factory C are $95\%$, $92\%$, and $96\%$, respectively.  
$(1)$ If Xiaozhang buys $1$ volleyball from the market, find the probability that the purchased volleyball is a qualified product.  
$(2)$ If Xiaoli goes to the market to wholesale $2$ volleyballs for resale, with $1$ volleyball from Factory A and $1$ volleyball from Factory C, and it is known that each volleyball from Factory A sold will earn a net profit of $10$ yuan, otherwise a loss of $5$ yuan, and the probability of selling each volleyball is equal to the qualification rate; each volleyball from Factory C sold will earn a net profit of $8$ yuan, otherwise a loss of $6$ yuan, and the probability of selling each volleyball is equal to the qualification rate. Find the mathematical expectation of the net profit Xiaoli will earn from wholesaling $2$ volleyballs in the market.

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