
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 completing the square is an important method in mathematics. It refers to the method of transforming a part of an expression into a perfect square or the sum of several perfect squares through equivalent transformations. 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 $11$ is not a "perfect number"; The number $53$ is a "perfect number." Therefore, the answers are: not, yes.

$(2)$ Given $x^{2}+y^{2}-4x+2y+5=0$, then $x+y=1$.

$(3)$ When $k=36$, $S$ is a "perfect number," the reason is as follows: $S=2x^{2}+y^{2}+2xy+12x+k$ $=(x^{2}+12x+k)+(y^{2}+2xy+x^{2})$ $=(x^{2}+12x+k)+(y+x)^{2}$, because $S$ is a perfect number, therefore $x^{2}+12x+k$ is a perfect square, therefore $k=36$.

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