
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 transforming a quadratic trinomial (or part of it) of the form $ax^{2}+bx+c$ into a perfect square trinomial is called the method of completing the square. The basic form of completing the square is the reverse of the perfect square formula, that is, $a^{2}\pm 2ab+b^{2}=\left(a\pm b\right)^{2}$. For example, the form of $x^{2}-2x+4=\left(x-1\right)^{2}+3$. We define: an integer that can be expressed in the form $a^{2}+b^{2}$ (where $a$ and $b$ are integers) is called a "perfect number". For example, $10$ is a "perfect number" because $10=3^{2}+1^{2}$.

Solve the problem:
$(1)$ Among the following numbers, the "perfect numbers" are ______ (fill in the serial number).
① $29$; ② $48$; ③ $13$; ④ $28$.

Investigate the problem:
$(2)$ If $a^{2}-4a+8=\left(a-m\right)^{2}+n^{2}$ (where $m$ and $n$ are constants), then the value of $mn$ is ______;
$(3)$ Given $S=a^{2}+4ab+5b^{2}-12b+k$ (where $a$ and $b$ are integers, and $k$ is a constant), when $k=$ ______, $S$ is a "perfect number".

Expand the application:
$(4)$ Given real numbers $a$ and $b$ satisfy $-a^{2}+5a+b-7=0$, then the minimum value of $a+b$ is ______.

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