
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 $C_1: x^2+4y^2=1$, with the foci on the x-axis, the length of the minor axis of ellipse $C_2$ is equal to the length of the major axis of $C_1$, and its eccentricity is $\frac{\sqrt{3}}{2}$.  
(1) Find the equation of ellipse $C_2$;  
(2) If point $T$ satisfies $\overrightarrow{OT} = \overrightarrow{MN} + 2\overrightarrow{OM} + \overrightarrow{ON}$, where $M$ and $N$ are points on $C_2$, and the product of the slopes of lines $OM$ and $ON$ equals $-\frac{1}{4}$, does there exist two fixed points $A$ and $B$ such that $|TA|+|TB|$ is a constant? If so, find this constant; if not, explain why.

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