
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 that the line $l_{1}$ is the tangent to the curve $y=x^{2}+x-2$ at the point $(1,0)$, and $l_{2}$ is another tangent line to the curve such that $l_{1} \perp l_{2}$.

$(1)$ Find the equation of the line $l_{2}$;  
$(2)$ Find the area of the triangle formed by the lines $l_{1}$, $l_{2}$, and the $x$-axis.

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