
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 the following material and answer the questions:<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 the method of completing the square. Completing the square is an important mathematical method for solving problems. It can not only factorize a seemingly indivisible polynomial, but also solve some problems related to non-negative numbers or find the maximum and minimum values of algebraic expressions. For example, to factorize $x^{2}+2x-3$, we can perform the following operations: $x^{2}+2x-3=(x^{2}+2x+1)-4=(x+1)^{2}-4$, then using the difference of squares formula, we get $x^{2}+2x-3=(x+3)(x-1)$; another example: to find the minimum value of the algebraic expression $2x^{2}+4x-6$, we can transform the expression as follows: $2x^{2}+4x-6=2(x^{2}+2x-3)=2(x+1)^{2}-8$, so by the non-negativity of squares, when $x=-1$, $2x^{2}+4x-6$ has a minimum value of $-8$. According to the reading material, solve the following problems using the method of completing the square:<br/>$(1)$ If the polynomial $x^{2}-4x+k$ is a perfect square trinomial, then the constant $k=\_\_\_\_\_\_.$<br/>$(2)$ Factorize the expression: $x^{2}-4x-12=$______, and the minimum value of the algebraic expression $2x^{2}-8x-24$ is______.<br/>$(3)$ Determine which is greater between the algebraic expressions $a^{2}+2b^{2}+11$ and $2ab+2a+4b$, and explain the reason.

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