
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 curves $C_1$ and $C_2$ with the following parameterized equations:

$C_1$: $\begin{cases} x = -4 + \cos t \\ y = 3 + \sin t \end{cases}$ ($t$ is a parameter)

$C_2$: $\begin{cases} x = 8 \cos \theta \\ y = 3 \sin \theta \end{cases}$ ($\theta$ is a parameter)

(1) Convert the equations of $C_1$ and $C_2$ to standard form and explain the type of curve they each represent.

(2) Let $P$ be a point on curve $C_1$ corresponding to the parameter value $t=\frac{\pi}{2}$, and let $Q$ be a moving point on curve $C_2$. Determine the position of $Q$ such that the distance between the midpoint $M$ of $PQ$ and the line $C_3$: $\begin{cases} x = 3 + 2t \\ y = -2 + t \end{cases}$ ($t$ is a parameter) is minimized, and find the minimum distance.

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