
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
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 as 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 |

$(1)$ The length of the spring when no object is hung on it is ______ $cm$;<br/>
$(2)$ When the mass of the object hung is $5kg$, the length of the spring is ______ $cm$;<br/>
$(3)$ Within the elastic limit, if the mass of the object hung is $x$ $(kg)$ and the length of the spring is $y$ $(cm)$, write the relationship between $y$ and $x$ based on the table ______;<br/>
$(4)$ If the maximum length of the spring is $20cm$, then the spring can support a maximum mass of ______ $kg$.

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