
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 $M$ is a moving point on the circle $x^2+y^2=4$, and the positive projection of point $M$ on the $x$-axis is $H$. If $2\overrightarrow{HP}=\sqrt{3}\overrightarrow{HM}$,

 (Ⅰ) find the equation of the trajectory $C$ of point $P$;

 (Ⅱ) A line $l$ with slope $k$ ($k \neq 0$) is drawn through point $Q(-4,0)$, intersecting the trajectory $C$ at points $B$ and $D$, with $N$ being the midpoint of $BD$. Please explain that there exists a real number $k$ such that point $N$ lies on the circle $x^2+y^2=1$ (no need to find the real number $k$).

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