
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with a minor axis length of $2$, the line passing through the top vertex $E$ and the right focus $F$ is tangent to the circle $M$: $x^2 + y^2 - 4x - 2y + 4 = 0$.

$(I)$ Find the standard equation of the ellipse $C$;  
$(II)$ If the line $l$ passes through the point $(1,0)$ and intersects the ellipse $C$ at points $A$ and $B$, is there a point $T(t,0)$ $(t \neq 0)$ on the $x$-axis such that no matter how the slope of line $l$ changes, $\angle OTA = \angle OTB$ always holds (where $O$ is the origin)? If such a point exists, find the value of $t$; if not, explain why.

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