
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 an ellipse \(\frac{x^{2}}{a_{1}^{2}}+\frac{y^{2}}{b_{1}^{2}}=1\) \((a_{1}>b_{1}>0)\) and a hyperbola \(\frac{x^{2}}{a_{2}^{2}}-\frac{y^{2}}{b_{2}^{2}}=1\) \((a_{2}>0, b_{2}>0)\) with the same foci, let \(P\) be an intersection point of the two curves. The slopes of the tangents to the ellipse and hyperbola at point \(P\) are \(k_{1}\) and \(k_{2}\) respectively. What is the value of \(k_{1} k_{2}\)?

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