
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 fruit supermarket purchases a certain amount of grade $A$ oranges every day. The supermarket recorded the actual sales volume of oranges for $20$ days, and the statistical results are shown in the table below:

| Sales Volume (kg) | $150$ | $160$ | $170$ | $180$ | $190$ | $200$ |
|-------------------|-------|-------|-------|-------|-------|-------|
| Number of Days    | $3$   | $4$   | $6$   | $5$   | $1$   | $1$   |

$(1)$ Estimate the average and variance of the sales volume of grade $A$ oranges in the supermarket over these $20$ days.  
$(2)$ This year, the purchase price of grade $A$ oranges is $6$ yuan/kg, and the supermarket sells them at a price of $10$ yuan/kg. To ensure the quality of the oranges, if they are not sold on the same day, they are returned to the supplier at a price of $4$ yuan/kg. If the supermarket plans to purchase either $170$ kg or $180$ kg of oranges per day, do you think they should purchase $170$ kg or $180$ kg? Please explain your reasoning.

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