
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
The relationship between the number of cups of milk tea sold $x$ (cups) and the total sales price $y$ (yuan) in a certain milk tea shop is shown in the following table:

| Number of Milk Teas $x$ (cups) | $1$ | $2$ | $3$ | $4$ | $\ldots$ | $10$ | $\ldots$ |
|--------------------------------|-----|-----|-----|-----|----------|------|----------|
| Sales Total Price $y$ (yuan)   | $12$| $22$| $32$| $42$| $\ldots$ | $102$| $\ldots$ |

$(1)$ Write down the relationship between the total sales price $y$ (yuan) and the number of cups of milk tea $x$ (cups).  
$(2)$ If the shop wants to achieve a total sales price of $822$ yuan in one day, how many cups of milk tea need to be sold?

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