
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 $C\_2$, where $C\_1$: $\begin{cases} x = -4 + \cos t \\ y = 3 + \sin t \end{cases}$ ($t$ is a parameter) and $C\_2$: $\begin{cases} x = 6\cos \theta \\ y = 2\sin \theta \end{cases}$ ($\theta$ is a parameter).

1. Find the equations of $C\_1$ and $C\_2$ in the standard form and explain what type of curves they represent.
2. If the point $P$ on $C\_1$ corresponds to the parameter $t = \frac{\pi}{2}$, and $Q$ is a moving point on $C\_2$, find the minimum distance between the midpoint $M$ of $PQ$ and the line $C\_3$: $\begin{cases} x = -3\sqrt{3} + \sqrt{3}t \\ y = -3 - t \end{cases}$ ($t$ is a parameter).
3. If $Q$ is a moving point on the curve $C\_2$, find the minimum and maximum distances between $Q$ and the line $C\_3$.
4. If the point $P(x, y)$ is a moving point on the curve $C\_1$, find the range of values of $2x + y$.
5. If $x + y + a \geq 0$ is always true, where $(x, y)$ is on the curve $C\_1$, find the range of real values of $a$.

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