
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
The Fermat point is a point located inside a triangle such that the sum of its distances to the three vertices of the triangle is minimized. When all three interior angles of a triangle are less than $\frac{2\pi}{3}$, the angles formed by connecting the Fermat point with the three vertices of the triangle are all $\frac{2\pi}{3}$. Given that point $P$ is the Fermat point of triangle $ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, if $\cos A = 2\sin \left(C-\frac{\pi}{6}\right)\cos B$ and $b^{2} = \left(a-c\right)^{2}+6$, then the value of $PA \cdot PB + PB \cdot PC + PA \cdot PC$ is ____.

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