
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 an important method of thinking 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 identity 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 $a^{2}+b^{2}$ ($a$ 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 the Problems:**  
$(1)$ Given that $29$ is a "perfect number," please write it in the form of $a^{2}+b^{2}$ ($a$ and $b$ are integers).  
$(2)$ If $x^{2}-6x+5$ can be factored into $\left(x-m\right)^{2}+n$ ($m$ and $n$ are constants), then $mn=\_\_\_\_\_\_$.  
**Exploring the Problems:**  
$(3)$ Given $x^{2}+y^{2}-2x+4y+5=0$, find the value of $x+y$;  
$(4)$ Given $S=x^{2}+4y^{2}+4x-12y+k$ ($x$ and $y$ are integers, $k$ is a constant), to make $S$ a "perfect number," try to find a value of $k$ that satisfies the condition, and explain the reason.

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