
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
Given polynomials $A$ and $B$, and $2A+B=7ab+6a-2b-11$, $2B-A=4ab-3a-4b+18$.  
$(1)$ Reading material: We can always find polynomials $A$ and $B$ by adding parentheses, such as:  
$5B=\left(2A+B\right)+2\left(2B-A\right)$  
$=\left(7ab+6a-2b-11\right)+2\left(4ab-3a-4b+18\right)$  
$=15ab-10b+25$  
$\therefore B=3ab-2b+5$  
$(2)$ Application material: Please find the polynomial $A$ using a method similar to the reading material.  
$(3)$ Xiaohong substitutes a pair of values for $a$ and $b$ that are reciprocals into the polynomial $A$, and the value of $A$ is exactly 0. Find the value of the polynomial $B$.  
$(4)$ Smart Xiaogang found that as long as the letter $b$ takes a fixed number, regardless of what number the letter $a$ takes, the value of $B$ is always 7 greater than the value of $A$. What is the value of $b$ that Xiaogang took?

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