
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 two lines with slopes that are negative reciprocals of each other and intersect at point $P(1,1)$, the ratio of the lengths of the chords intercepted by the circle $O$: $({x}^{2}+{y}^{2}=4)$ is $\frac{\sqrt{6}}{2}$. Find the product of the slopes of these two lines.

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