
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
In a math textbook, it is written as follows:<br/>"We call the polynomials $a^{2}+2ab+b^{2}$ and $a^{2}-2ab+b^{2}$ as perfect square trinomials. If a polynomial is not a perfect square trinomial, we often perform the following transformation: first add a suitable term to make a perfect square trinomial appear in the expression, then subtract this term to keep the value of the entire expression unchanged. This method is called completing the square, which is an important mathematical method for solving problems, often used to solve problems related to non-negative numbers or to find the maximum or minimum values of algebraic expressions." For example: $x^{2}+2x-3=(x^{2}+2x+1)-4=\left(x+1\right)^{2}-4$; for example, find the minimum value of the algebraic expression $2x^{2}+4x-6$; $2x^{2}+4x-6=2(x^{2}+2x-3)=2\left(x+1\right)^{2}-8$. According to the reading material, solve the following problems using the method of completing the square:<br/>$(1)$ Factorize: $m^{2}-6m+5$ ______;<br/>$(2)$ For what values of $a$ and $b$ does the polynomial $a^{2}+b^{2}-4a+10b+33$ have a minimum value, and find this minimum value;<br/>$(3)$ Given $a-b=8$, $ab+c^{2}-4c+20=0$, find the value of $a+b+c$.

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