
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 coincides with the origin of the Cartesian coordinate system, and the polar axis coincides with the positive half-axis of the $x$-axis. If the equation of curve $C_1$ is $\rho^2 = 8\rho\sin\theta - 15$, and the equation of curve $C_2$ is 
\[
\begin{cases} 
x = 2\sqrt{2}\cos\alpha \\ 
y = \sqrt{2}\sin\alpha 
\end{cases}
\]
(where $\alpha$ is a parameter).

(1) Convert the equation of $C_1$ into a Cartesian coordinate equation;

(2) If the point $Q$ on $C_2$ corresponds to the parameter $\alpha = \frac{3\pi}{4}$, and $P$ is a moving point on $C_1$, find the minimum value of $PQ$.

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