
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
In a handicraft class, Xiaoming is preparing to make a kite in the shape of a rhombus. The sum of the lengths of the two diagonals of this rhombus is exactly $70$ cm. The area of the rhombus, $S$ (unit: $cm^{2}$), varies with the length of one of its diagonals, $x$ (unit: $cm$). <br/>$(1)$ Please write down the functional relationship between $S$ and $x$ directly (no need to specify the range of the independent variable $x$); <br/>$(2)$ When the area of the rhombus kite is $600 cm^{2}$, what is the length of the rhombus kite's side?

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