
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
A certain scientific research unit has developed a certain type of scientific research airship. Each airship of this type can perform up to $n$ $(n\in N^{*}$, $n\geqslant 2)$ scientific research missions. The airship performs the first scientific research mission with a success rate of $p$ $(0 \lt p \lt 1)$. If the $k$-th $(k=1,2,\ldots ,n-1)$ scientific research mission is successful, then the probability of successfully completing the $k+1$-th scientific research mission is also $p$. Otherwise, the airship will end the scientific research mission. Each time the airship of this type performs a scientific research mission, if it successfully returns, it can obtain scientific research data worth $X$ million yuan. The probability of "$X=0$" is $0.8$, and the probability of "$X=200$" is $0.2$. If it fails to return, no scientific research data can be obtained for that mission. Let $Y$ denote the total value of scientific research data that an airship of this type can obtain.

$(1)$ If $p=0.5$ and $n=2$, find the distribution of $Y$.

$(2)$ Find $E(Y)$ (expressed in terms of $n$ and $p$).

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