
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 a line $l$ with an inclination angle of $\alpha (\alpha \neq \frac{\pi}{2})$ is $\begin{cases} x=1+t\cos \alpha \\ y=t\sin \alpha \end{cases}$ (where $t$ is the parameter). In the polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho\cos^2 \theta - 4\sin \theta = 0$.

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

$(2)$ Given point $P(1,0)$. If the polar coordinates of point $M$ are $\left(1, \frac{\pi}{2}\right)$, and line $l$ passes through $M$ and intersects curve $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $Q$, find the value of $|PQ|$.

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