
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 vaccine production unit inspects the antibody production of a certain vaccine through blood tests. There are $n$ blood samples ($n\in \mathbb{N}^{*}$) available, with the following two inspection methods:<br/>Method 1: Inspecting one by one, requiring $n$ inspections;<br/>Method 2: Mixing inspections, combining $k$ blood samples ($k\in \mathbb{N}^{*}$ and $k\geqslant 2$) for inspection. If the mixed blood sample has no antibodies, it means that all $k$ blood samples have no antibodies, requiring only $1$ inspection. If the mixed blood sample has antibodies, in order to determine which specific blood sample has antibodies, each blood sample needs to be tested separately once, with a total of $\left(k+1\right)$ inspections. Assuming that the test results of each sample are independent of each other, the probability of each sample having antibodies is $p\left(0 \lt p \lt 1\right)$.<br/>$(1)$ There are currently $7$ different blood samples, with only $3$ blood samples containing antibodies. Using the one-by-one inspection method, find the probability of exactly $4$ inspections being able to detect all blood samples containing antibodies;<br/>$(2)$ Now take $k$ blood samples ($k\in \mathbb{N}^{*}$ and $k\geqslant 2$), denoting the total number of inspections required for the samples using the one-by-one inspection method as $\xi _{1}$; and using the mixing inspection method as $\xi _{2}$.<br/>① If $E(\xi _{1})=E(\xi _{2})$, find the function relationship $p=f\left(k\right)$ with respect to $k$;<br/>② Given $p=1-{e}^{-\frac{1}{8}}$, based on the expected total number of inspections, discuss which inspection method is better?<br/>Reference data: $\ln 2=0.693$, $\ln 25=3.219$, $\ln 26=3.258$, $\ln 27=3.296$, $\ln 28=3.332$.

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