
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
[Project Study] Factoring 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. For example, factor the quadratic trinomial $x^{2}-2x+3$.

Solution: $x^{2}-2x+3=x^{2}-2x+1+2=(x^{2}-2x+1)+2=(x-1)^{2}+2$. 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" because $5=2^{2}+1^{2}$. Another example is $M=x^{2}+2xy+2y^{2}=\left(x+y\right)^{2}+y^{2}$ (where $x$ and $y$ are integers), so $M$ is also a "perfect number."

[Problem Solving]

1. Among the following numbers, the "perfect numbers" are:
   ① $10$
   ② $45$
   ③ $28$
   ④ $29$

2. If the quadratic trinomial $x^{2}-6x+13$ (where $x$ is an integer) is a "perfect number" and can be factored as $\left(x-m\right)^{2}+n$ (where $m$ and $n$ are constants), then the value of $mn$ is ______.

[Problem Exploration]

3. Given $S=x^{2}+9y^{2}+8x-12y+k$ (where $x$ and $y$ are integers, and $k$ is a constant), to make $S$ a "perfect number," find the values of $k$ that satisfy the condition.

[Problem Extension]

4. Given real numbers $x$ and $y$ satisfying $-x^{2}+7x+y-10=0$, find the minimum value of $x+y$.

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