
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
Read first, then solve. Given $m^{2}+2m+n^{2}-6n+10=0$, find the values of $m$ and $n$. Solution: The equation can be transformed into $(m^{2}+2m+1)+(n^{2}-6n+9)=0$. That is, $\left(m+1\right)^{2}+\left(n-3\right)^{2}=0$. Since $\left(m+1\right)^{2}\geqslant 0$ and $\left(n-3\right)^{2}\geqslant 0$, we have $m+1=0$ and $n-3=0$, which leads to $m=-1$ and $n=3$. The method of transforming algebraic expressions to make them appear in complete square form is called "completing the square". Please use this method to solve the following problems: (1) Given $a$ and $b$ are the length and width of rectangle $ABCD$, satisfying $a^{2}+b^{2}-8a-6b+25=0$, then the area of rectangle $ABCD$ is ______; (2) Find the minimum value of the algebraic expression $a^{2}+4b^{2}+4ab-4a-8b+7$, and determine the relationship between $a$ and $b$ at that time; (3) Compare the polynomials $x^{2}+3x-4$ and $2x^{2}+2x-3$, and explain the reasoning.

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