
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 $xoy$, a line passes through point $P(1,0)$, and its inclination angle $\alpha$ is obtuse. With the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho^2(1+2\sin^2\theta)=3$.

(1) Write the parametric equation of line $l$ and the Cartesian equation of curve $C$;

(2) If $\alpha =\frac{5\pi }{6}$, line $l$ intersects curve $C$ at two distinct points $M$ and $N$. Find the length of $|MN|$.

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