
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
The method of pairing is one of the important mathematical methods. It refers to the method of transforming a part of an expression into a sum of one or more complete square forms 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 represented in the form of $a^{2}+b^{2}$ ($a$ and $b$ are integers) is called a "perfect number". For example, $5$ is a "perfect number" because $5=2^{2}+1^{2}$. 

**Solving the Problems:**

$(1)$ Given that $13$ is a "perfect number", express it in the form of $a^{2}+b^{2}$ where $a$ and $b$ are positive integers. 

$(2)$ If $x^{2}-4x+53$ can be factored into $\left(x-m\right)^{2}+n^{2}$ where $m$ and $n$ are positive integers, then find the value of $m+n$.

**Exploring the Problem:**

$(3)$ Given $S=x^{2}+9y^{2}+8x-12y+k$ where $x$ and $y$ are integers and $k$ is a constant. In order for $S$ to be a "perfect number", find a value of $k$ that satisfies the condition and explain the reasoning.

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