
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 tree with black pearls hidden, this item is only available in May. Travelers who pass by taste one, with a mouthful of sweetness and sourness, never wanting to leave." The Dongkui waxberry is a sweet gift in summer. Each batch of Dongkui waxberries must undergo two rounds of testing before entering the market. They can only be sold if they pass both rounds of testing; otherwise, they cannot be sold. It is known that the probability of not passing the first round of testing is $\frac{1}{9}$, and the probability of not passing the second round of testing is $\frac{1}{10}$. The two rounds of testing are independent of each other.<br/>$(1)$ Find the probability that a batch of waxberries cannot be sold;<br/>$(2)$ If the waxberries can be sold, the profit for that batch is $400$ yuan; if the waxberries cannot be sold, the batch will incur a loss of $800$ yuan (i.e., a profit of $-800$ yuan). It is known that there are currently 4 batches of waxberries. Let $X$ represent the profit from the 4 batches of waxberries (the sales of waxberries in each batch are independent of each other). Find the probability distribution and mathematical expectation of $X$.

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