
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 a very important mathematical thinking method, which refers to the method of transforming an expression or a part of an expression into a complete square or the sum of several complete squares through constant transformation.<br/>Definition: If an integer can be expressed in the form of $a^{2}+b^{2}$ ($a$, $b$ are integers), then this number is called a "perfect number".<br/>For example, $5$ is a "perfect number". Reason: because $5=1^{2}+2^{2}$, so $5$ is a "perfect number".<br/>Solving the problem:<br/>$(1)$ Given that $29$ is a "perfect number", please write it in the form of $a^{2}+b^{2}$ ($a$, $b$ are integers): ______;<br/>$(2)$ Similarly, an entire expression expressed in the form of $M^{2}+N^{2}$ ($M$, $N$ are entire expressions) is called a "perfect expression". If $x^{2}-4x+5$ can be paired as $\left(x-m\right)^{2}+n$ ($m$, $n$ are constants), then the value of $mn$ is ______;<br/>Exploring the problem: Given that $x^{2}+y^{2}-2x+4y+5$ is a "perfect expression", if $x^{2}+y^{2}-2x+4y+5=0$, find the value of $x+y$.

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