
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 bread lover. 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 means that 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 $k$ ($k\in \mathbb{N}^*$, $k\geqslant 2$) data points from the values of $X$, 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.  
① Assuming the baker's claim is true, randomly purchase $25$ loaves of bread. Let the average weight of these $25$ loaves be $Y$. Find $P(Y < 980)$.  
② Poincaré weighs and records the bread he buys every day. After 25 days, all the recorded weights fall within $(950, 1050)$, and the average weight of the 25 loaves is calculated to be $978.72g$. Poincaré reports the baker based on a probability analysis. Explain why Poincaré reported the baker from a probability perspective.  

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

Attachment:  
① If a random variable $\eta$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu - \sigma \leq \eta \leq \mu + \sigma) = 0.6827$, $P(\mu - 2\sigma \leq \eta \leq \mu + 2\sigma) = 0.9545$, $P(\mu - 3\sigma \leq \eta \leq \mu + 3\sigma) = 0.9973$.  
② Events with a probability less than $0.05$ are usually referred to as small probability events, which are unlikely to occur.

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