
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 squares through constant 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 $a^{2}+b^{2}$ ($a$ and $b$ are positive 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 $29$ is a "perfect number", please express it in the form of $a^{2}+b^{2}$ ($a$ and $b$ are positive integers) as ______.
2. If $x^{2}-6x+13$ can be paired into $\left(x-m\right)^{2}+n^{2}$ ($m$ and $n$ are positive integers), then $mn=$ ______.

**Exploring the Problem:**

3. Given $S=x^{2}+4y^{2}+4x-12y+k$ ($x$ and $y$ are integers, $k$ is a constant), in order to make $S$ a "perfect number", 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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