
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 an ellipse $T:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$, where the slope of the line $l_{1}$ is $-\frac{1}{2}$ and it intersects the ellipse $T$ at only one point $P(1,\frac{3}{2})$.<br/>$(1)$ Find the standard equation of the ellipse $T$;<br/>$(2)$ A line passing through the right focus $F$ of the ellipse and intersecting the ellipse $T$ at points $A$ and $B$, with point $C$ lying on the line $l_{2}: x=4$, and $BC$ parallel to the $x$-axis. Find the intercept of the line $AC$ on the $x$-axis.

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