
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
[Reading Material] The method of pairing is an important thinking method in mathematics. It refers to the method of transforming a part of an expression into a complete square or the sum of several complete squares through constant transformation. This method is often used in the transformation of algebraic expressions and combined with the meaning of non-negative numbers to solve some problems. We define: an integer that can be expressed in the form of $a^{2}+b^{2}$ (where $a$ and $b$ are integers) is called a "perfect number." For example, $5$ is a "perfect number." Reason: because $5=2^{2}+1^{2}$, so $5$ is a "perfect number."

[Solving Problems]
$(1)$ The number $53$ is ______ a "perfect number" (fill in "is" or "is not").
$(2)$ Given $x^{2}+y^{2}-4x+2y+5=0$, then $x+y=$ ______.
$(3)$ Given $S=2x^{2}+y^{2}+2xy+12x+k$ (where $x$ and $y$ are integers, $k$ is a constant), to make $S$ a "perfect number," find the values of $k$ that satisfy the condition, and explain the reason.
$(4)$ Given real numbers $x$ and $y$ satisfy $-x^2+\frac{7}{2}x+y-3=0$, find the maximum value of $x-2y$.

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