
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
Most countries in the world use the Celsius temperature scale to forecast the weather, but countries like the United States and the United Kingdom still use the Fahrenheit temperature scale. A student looked up data and found the following information in the table:

| Celsius Temperature $x/^{\circ}\mathrm{C}$ | $0$ | $10$ | $20$ | $30$ | $40$ | $50$ |
|-------------------------------------------|-----|------|------|------|------|------|
| Fahrenheit Temperature $y/^{\circ}F$       | $32$| $50$ | $68$ | $86$ | $104$| $122$|

$(1)$ Analyze whether the correspondence between the two temperature scales is a linear function. Fill in the blank with "yes" or "no".  
$(2)$ Based on the data, calculate the Celsius temperature at $0^{\circ}F$.

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