
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 a line $l$ intersects with the ellipse $\displaystyle {x^{2}}+2{y^{2}}=2$ at points $P_{1}$ and $P_{2}$, the midpoint of the segment $P_{1}P_{2}$ is $P$, and $O$ is the origin. Suppose the slope of line $l$ is $k_{1} (k_{1} \neq 0)$, and the slope of line $OP$ is $k_{2}$, find the value of $k_{1}k_{2}$.

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