
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 $C_1$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ (where $a>b>0$) and a hyperbola $C_2$: $\frac{{x}^{2}}{4}-y^{2}=1$ have two coincident vertices, and the distance from the focus of $C_2$ to its asymptote is exactly the length of the short axis of $C_1$.  
(1) Find the standard equation of the ellipse $C_1$;  
(2) Given a point $T(t,0)$ where $t\in(-a,0)\cup(0,a)$, a non-vertical line $l$ passing through $T$ intersects $C_1$ at points $A$ and $B$. Does there exist a point $M$ on the x-axis such that $MT$ bisects $\angle AMB$? If so, find the coordinates of point $M$; if not, explain why.

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