
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 line $l$ passes through the point $M(-1,m)$ with a slope of $1$. A polar coordinate system is established with the origin as the pole and the positive $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho \sin ^{2}\theta =2p\cos \theta (p \gt 0)$. The line $l$ intersects the curve $C$ at two distinct points $A$ and $B$.
- $(1)$ Write down a parametric equation for the line $l$, and find the Cartesian equation of the curve $C$.
- $(2)$ If point $M$ lies on the focal chord of curve $C$, and $|MA|$, $\frac{1}{2}|AB|$, and $|MB|$ form a geometric sequence, find the value of $m$.

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