
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, a polar coordinate system is established with the origin as the pole and the positive half-axis of the x-axis as the polar axis. It is known that the polar coordinate equation of curve $C$ is $\rho \sin ^{2}\theta =a\cos \theta \left(a \gt 0\right)$. The line $l$ passing through the point $P(-2,-4)$ has the parameter equation $\left\{\begin{array}{l}x=-2+\dfrac{\sqrt{2}}{2}t\\y=-4+\dfrac{\sqrt{2}}{2}t\end{array}\right. (t$ is the parameter), and the line $l$ intersects the curve $C$ at points $A$ and $B$.

1. Write down the rectangular coordinate equation of curve $C$ and the standard equation of line $l$.
2. If $|PA|\cdot |PB|=|AB|^{2}$, find the value of $a$.

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