
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 recently introduced a batch of passion fruits, priced at $8$ yuan per jin. In order to set a reasonable price, a flexible pricing strategy was implemented in the first week. The selling price per jin was set at $10$ yuan as the standard. Any amount exceeding $10$ yuan was recorded as positive, and any amount less than $10$ yuan was recorded as negative. The supermarket recorded the selling price and quantity of passion fruits for the first week as follows:

| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|-----------------|--------|---------|-----------|----------|--------|----------|--------|
| Price relative to standard price (yuan) | $+1$ | $-2$ | $+3$ | $-1$ | $+2$ | $+5$ | $-4$ |
| Quantity sold (jin) | $20$ | $35$ | $10$ | $30$ | $15$ | $5$ | $50$ |

$(1)$ The highest unit price of passion fruits sold by the supermarket in that week was on ________ and the highest price was ________ yuan.  
$(2)$ How was the profit/loss of the supermarket from selling this type of passion fruit during the week?  
$(3)$ In order to promote this type of passion fruit, the supermarket decided to introduce two promotional methods starting from the next week:  
Method one: Buy up to $5$ jin of passion fruit at $12$ yuan per jin, and receive a $20\%$ discount for any amount exceeding $5$ jin.  
Method two: Selling price of $10$ yuan per jin.  
A certain unit decided to buy $35$ jin of passion fruit. By calculation, determine which method is more cost-effective.

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