
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 ratio of the length of the tangent line from point P to the circle $(x+2)^2 + y^2 = 1$ to the distance from P to the y-axis is $t (t > 0, t \neq 1)$,

(1) Find the equation of the locus C of the moving point P;

(2) When $t = \sqrt{3}$, move the graph of the locus C to the left by 1 unit along the x-axis to obtain the curve G. Through a point Q on the curve G, draw two vertical lines to the feet $P_1$ and $P_2$. Find the value of $\overrightarrow{QP_1} \cdot \overrightarrow{QP_2}$;

(3) Let $F_1$ and $F_2$ be the two foci of the curve C. Find the range of t such that there is no point Q on the curve C satisfying $\angle F_1QF_2 = \theta (0 < \theta < \pi)$.

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