
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
Explore 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 hanging object (in kg) is shown in the table below:

| Mass of Hanging 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  |

$(1)$ When the mass of the hanging object is $3$ kg, the length of the spring is ________ cm.  
$(2)$ Within the elastic limit, if the mass of the hanging object is $x$ kg and the length of the spring is $y$ cm, write the relationship between $y$ and $x$ based on the table.  
$(3)$ When the mass of the hanging object is $5.5$ kg, find the length of the spring.  
$(4)$ If the maximum length of the spring is $20$ cm, what is the maximum mass of the object that can be hung on the spring?

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