
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 curve $C$ is
$$
\begin{cases}
x = 2t^2 - 1\\
y = 2t - 1
\end{cases}
(t \text{ is the parameter}).
$$
A polar coordinate system is established with the origin of the Cartesian coordinate system as the pole and the positive $x$-axis as the polar axis. It is known that the polar equation of line $l$ is $\rho(2\sin \theta - \cos \theta) = m$.

$(1)$ Find the general equation of curve $C$.

$(2)$ If line $l$ is tangent to curve $C$ and intersects the coordinate axes at points $A$ and $B$, find the equation in rectangular coordinates for the circle with diameter $AB$.

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