
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 the parabola $y=x^{2}+bx-2$ passes through the point $(1,3)$, intersects the $y$-axis at point $A$, and has its vertex at point $B$. Let $k$ be the abscissa of the intersection point of the parabola $y=x^{2}+bx-2$ with the $x$-axis.  
$(1)$ Find the area of $\triangle OAB$;  
$(2)$ Find the value of the algebraic expression $\frac{4k^4+3k^2+12k-6}{k^8+2k^6+k^5-2k^3+8k^2+16}$.

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