
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 a segment AB of length 2 with endpoints A and B sliding on the x-axis and y-axis respectively, the trajectory of the midpoint M of segment AB is curve C.
(Ⅰ) Find the equation of curve C;
(Ⅱ) If point P (x, y) is a moving point on curve C, find the range of values for 3x-4y;
(Ⅲ) Given a fixed point Q (0, $\frac{2}{3}$), investigate whether there exists a fixed point T (0, t) (t $\neq \frac{2}{3}$) and a constant $\lambda$ such that for any point S on curve C, $|ST| = \lambda|SQ|$ holds? If it exists, find t and $\lambda$; if not, explain why.

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