
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 the ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{1}{2}$, and the point $(1, \frac{3}{2})$ lies on the ellipse $E$.

(I) Find the equation of the ellipse $E$;

(II) Let the moving line $l$: $y=kx+m(k,m∈R)$ intersect with the ellipse $E$ at only one point $P$.

(1) Express the coordinates of point $P$ using real numbers $k$ and $m$;

(2) If the moving line $l$ intersects with the line $x=4$ at point $Q$, is there a fixed point $M$ on the $x$-axis such that $MP⊥MQ$? If it exists, find the coordinates of the fixed point $M$; if not, explain the reason.

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