
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
In the Cartesian coordinate plane, the coordinates of two vertices of $\triangle ABC$ are $A(- \frac { \sqrt {7}}{7}a，0)$, $B( \frac { \sqrt {7}}{7}a，0)$ ($a＞0$), and two moving points M, N satisfy $\overrightarrow {MA}+ \overrightarrow {MB}+ \overrightarrow {MC}= \overrightarrow {0}$, $| \overrightarrow {NC}|= \sqrt {7}| \overrightarrow {NA}|= \sqrt {7}| \overrightarrow {NB}|$, and the vector $\overrightarrow {MN}$ is collinear with $\overrightarrow {AB}$.  
(1) Find the trajectory equation of vertex C of $\triangle ABC$;  
(2) If the line passing through point P(0, a) intersects the trajectory from (1) at points E and F, find the range of values for $\overrightarrow {PE}\cdot \overrightarrow {PF}$.  
(3) If G(-a, 0), H(2a, 0), and $\theta$ is any point on the trajectory of point C in the first quadrant, is there a constant $\lambda$ ($\lambda＞0$) such that $\angle QHG=\lambda\angle QGH$ always holds? If it exists, find the value of $\lambda$; if not, please explain why.

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