
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
There are three different types of fruit tree seedlings $A$, $B$, $C$. After the introduction experiment, it was found that the natural survival rate of seedling $A$ is $0.6$, and the natural survival rate of seedlings $B$ and $C$ is both $p$ $(0.6 \leq p \leq 0.8)$.  
$(1)$ Take one seedling each of $A$, $B$, and $C$. Let $X$ be the number of seedlings that survive naturally. Find the probability distribution and $E(X)$.  
$(2)$ Take the value of $p$ that minimizes $E(X)$ from part (1) as the probability of natural survival for seedling $B$. The farmer decides to introduce $n$ ($n \in \mathbb{N}^{*}$) seedlings of type $B$. After introduction, 80% of the seedlings that do not survive naturally can be treated with artificial cultivation techniques, with a survival probability of $0.5$, while the remaining seedlings cannot survive.  
① Find the final survival probability of one seedling of type $B$.  
② If the profit from each surviving seedling is $400, and the loss from each non-surviving seedling is $60, the farmer wants to make a profit of at least $300,000. How many seedlings of type $B$ should be introduced at least?

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