
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
Exploring Calculation: When an object is hung on a spring, the spring will stretch. The relationship between the length of a spring (in cm) and the mass of the object hung on it (in kg) is shown in the table below:

| Mass of Object (kg) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---------------------|---|---|---|---|---|---|---|---|
| Length of Spring (cm) | 12 | 12.5 | 13 | 13.5 | 14 | 14.5 | 15 | 15.5 |

$(1)$ Within the elastic limit, if the mass of the object hung on the spring is $x$ kg and the length of the spring is $y$ cm, write the relationship between $y$ and $x$ based on the table above.  
$(2)$ If the maximum length of the spring is $20$ cm, what is the maximum mass of an object that can be hung on the spring?

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