
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 point H moves on the circle D: $(x-2)^2+(y+3)^2=32$, and point P has coordinates $(-6,3)$. The midpoint of the line segment PH is M.
(1) Find the equation of the trajectory of point M;
(2) Determine whether there is a fixed point A $(a,b)$ $(a \neq 0)$ in the plane such that $|MO| = \lambda |MA|$ (where $\lambda \neq 1$ is a constant). If such a point exists, find the coordinates of A and the value of $\lambda$. If not, explain why;
(3) If the line $y=kx$ intersects the path of M at points B and C, and point N $(0,t)$ is such that $NB \perp NC$, find the range of the real number $t$.

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