
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
Triangle A has a perimeter of $3a^{2}-6b+8$, and triangle B has a first side length of $a^{2}-2b$, a second side length of $a^{2}-3b$, and a third side length that is $a^{2}-2b-5$ shorter than the second side.<br/>$(1)$ Find the length of the third side of triangle B;<br/>$(2)$ Which triangle, A or B, has a greater perimeter? Explain your reasoning;<br/>$(3)$ Given that $a$ and $b$ are both positive integers, the points on the number line representing the perimeters of triangles A and B are denoted as points $A$ and $B$ respectively. If there are exactly 18 "integer points" (points representing integers) between points $A$ and $B$, find the value of $a$.

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