
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 $y^{2}=-x$ and the line $l$: $y=k(x+1)$ intersect at points $A$ and $B$, with point $O$ being the origin.
$(1)$ Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$;  
$(2)$ If the area of $\triangle OAB$ is equal to $\frac{5}{4}$, find the equation of line $l$.

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