
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 \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) where \(a, b \in \mathbb{R}^{+}\), with the semi-focal distance denoted by \(c\), and \(b^{2} = a c\). Let \(P\) and \(Q\) be any two points on the hyperbola, and let \(M\) be the midpoint of \(PQ\). When the slopes \(k_{PQ}\) of \(PQ\) and \(k_{OM}\) of \(OM\) exist, find the value of \(k_{PQ} \cdot k_{OM}\).

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