
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
In the Cartesian coordinate system $xOy$, the parametric equation of line $C_1$ is $\begin{cases} & x=1+\frac{1}{2}t \\ & y=\frac{\sqrt{3}}{2}t \end{cases}$ ($t$ is the parameter), and in the polar coordinate system with the origin as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $\rho^2(1+2\sin^2\theta)=3$.

$(1)$ Write the general equation of $C_1$ and the Cartesian coordinate equation of $C_2$;

$(2)$ Line $C_1$ intersects curve $C_2$ at points $A$ and $B$, with point $M(1,0)$. Find $||MA|-|MB||$.

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