
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 ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ passing through point $P(1, \frac{3}{2})$, with eccentricity $e= \frac{1}{2}$.
(I) Find the equation of ellipse $C$.
(II) Given line $l$: $x=my+1$ intersects the ellipse at points $A$ and $B$, let $t$ be the product of the slopes of the three lines containing the sides of $\triangle ABP$, find the maximum value of $t$.

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