
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 that the pole of the polar coordinate system is at the origin of the rectangular coordinate system, and the polar axis coincides with the non-negative semi-axis of the $x$-axis, the polar coordinate equation of line $l$ is $3\rho \cos \theta + \rho \sin \theta - 6 = 0$, and the parametric equation of circle $C$ is $\begin{cases} x = \sqrt{5}\cos \alpha \\ y = 1 + \sqrt{5} \sin \alpha \end{cases}$.

(1) Find the length of the chord cut off by the line on the circle.
(2) Given point $P(0, -2)$, let $l'$ be the line passing through $P$ and intersecting the circle at two distinct points $A$ and $B$. Find the absolute value of the dot product of vectors $\overrightarrow{PA}$ and $\overrightarrow{PB}$.

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