
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 three vertices of $\triangle ABC$ are all on the hyperbola $\frac{x^{2}}{2} - \frac{y^{2}}{4} = 1$, the slopes of the three sides $AB$, $BC$, $AC$ all exist and are non-zero, and their sum is $-1$; furthermore, the midpoints of $AB$, $BC$, $AC$ are $M$, $N$, $P$ respectively, and $O$ is the coordinate origin. The slopes of the lines $OM$, $ON$, $OP$ are $k_{1}$, $k_{2}$, $k_{3}$ respectively, and are all non-zero. Find $\frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}} = \_\_\_\_\_\_$.

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