
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 hyperbola $C: x^{2}- \frac{2y^{2}}{\sqrt{5}+1}=1$, and suppose a line $l$ passes through the center of $C$ and intersects $C$ at two points $M$ and $N$. Let $P$ be any point on the curve $C$. If the slopes of the lines $PM$ and $PN$ both exist and are denoted by $k_{PM}$ and $k_{PN}$ respectively, then find the value of $k_{PM} \cdot k_{PN}$.

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