
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
To consolidate and expand the achievements of poverty alleviation, and comprehensively promote rural revitalization, with the strong support of the national industrial poverty alleviation policy, a certain impoverished village is using local natural conditions to plant apples on the south and north mountains. They have now started to produce a large number of apples. When the apples are ripe, they are classified into "first grade," "second grade," and "third grade," with prices from high to low. A fruit merchant wants to purchase apples from here. Before purchasing, 200 kilograms of apples were randomly picked from the south and north mountains. After being separated by grade, the data obtained are as follows: on the south mountain, there are 40 kilograms of "first grade" apples and 150 kilograms of "second grade" apples; on both the south and north mountains, there are a total of 40 kilograms of "third grade" apples; on the north mountain, there are 50 kilograms of "first grade" apples. (Assuming that the total yield of apples on the two mountains is the same, estimate the probability based on the frequency of the sample)

$(1)$ If the cost of planting apples is $5$ yuan per kilogram, and the apple purchase prices are as shown in the table below:

| Grade | "First Grade" | "Second Grade" | "Third Grade" |
|-------|---------------|----------------|---------------|
| Price (yuan per kilogram) | $12$ | $8$ | $1$ |

① Calculate the average profit of the 200 kilograms of apples randomly picked from the south and north mountains separately;

② If calculated by quantity, there are on average 3 apples per kilogram of "first grade" apples, 4 apples per kilogram of "second grade" apples, and 6 apples per kilogram of "third grade" apples. Calculate the number of apples in the 200 kilograms of apples on the south mountain, and randomly select 13 apples from them using stratified sampling based on the number of apples of each grade. Place these 13 apples in 13 packages that look exactly the same. The fruit merchant randomly picks 2 apples from these 13 apples. Find the probability that exactly 1 "third grade" apple is picked.

$(2)$ Determine whether it can be said with 99% confidence that the amount of "third grade" apples is related to the south and north mountains.

Given: ${K}^{2}=\frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)},n=a+b+c+d$.

| $P(K^{2}\geqslant k₀)$ | $0.1$ | $0.05$ | $0.01$ | $0.005$ |
|------------------------|-------|--------|--------|---------|
| $k$₀ | $2.706$ | $3.841$ | $6.635$ | $7.879$ |

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