
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
French mathematician Poincaré is a person who likes to eat bread. He goes to the same bakery every day to buy a loaf of bread. The baker at the bakery claims that the average weight of the bread he sells is $1000g$, with a fluctuation of no more than $50g$. In mathematical terms, this statement can be expressed as: the weight of each loaf of bread follows a normal distribution with an expectation of $1000g$ and a standard deviation of $50g. 

$(1)$ Given the following conclusion: If $X\sim N(\mu, \sigma^2)$, randomly select $k$ data points from the values of $X$ ($k\in \mathbb{N}^*, k\geq 2$), and denote the average of these $k$ data points as $Y$, then the random variable $Y$ follows $N(\mu, \frac{{\sigma^2}}{k})$. Use this conclusion to solve the following problems: 

$(i)$ Assuming the baker's claim is true, randomly purchase $25$ loaves of bread. Let the average weight of these $25$ loaves be $Y$, find $P(Y\leq 980)$; 

$(ii)$ Poincaré weighs and records the bread he buys every day. After $25$ days, all the data fall within $(950, 1050)$, and the calculated average weight of the $25$ loaves is $978.72g$. Poincaré reported the baker based on this data. Explain from a probability perspective why Poincaré reported the baker; 

$(2)$ Assuming there are two identical boxes containing bread (except for the color, everything else is the same), it is known that the first box contains a total of $6$ loaves of bread, with $2$ black loaves; the second box contains a total of $8 loaves of bread, with $3$ black loaves. Now, randomly select a box, and then randomly pick $2$ loaves of bread from that box. Find the distribution table of the number of black loaves drawn and the mathematical expectation. 

Given: 
$(1)$ If a random variable $\eta$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu -\sigma \leq \eta \leq \mu +\sigma) = 0.8627$, $P(\mu -2\sigma \leq \eta \leq \mu +2\sigma) = 0.9545$, $P(\mu -3\sigma \leq \eta \leq \mu +3\sigma) = 0.9973$; 
$(2)$ Events with a probability less than $0.05$ are usually referred to as small probability events, which are unlikely to occur.

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