
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 parametric equation of line $l$: 
$$
\begin{cases}
x = -4t + a \\
y = 3t - 1
\end{cases}
$$
(where $t$ is the parameter), in the Cartesian coordinate system $xOy$, taking point O as the pole and the non-negative half-axis of the x-axis as the polar axis, with the same unit of length, a polar coordinate system is established. Suppose the equation of circle M is $\rho^2 - 6\rho \sin \theta = -8$.  
(Ⅰ) Find the Cartesian coordinate equation of circle M;  
(Ⅱ) If the chord intercepted by line $l$ on circle M has length $\sqrt{3}$, determine the value of the real number $a$.

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