
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
A line l passes through point A (-3, -$$\frac{3}{2}$$) with an inclination angle of α intersects with curve C: 
$$
\begin{cases}
x = 5\cos\theta \\
y = 5\sin\theta
\end{cases} \quad (\theta \text{ is a parameter})
$$
at points B and C.

(1) Write the parametric equation of the line l and find the length of the chord BC when α = $$\frac{\pi}{6}$$.
(2) Find the equation of line BC when A is exactly the midpoint of BC.
(3) Find the equation of line BC when |BC| = 8.
(4) As α varies, find the equation of the trajectory of the midpoint of chord BC.

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