
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 rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $\begin{cases} x=-\frac{1}{2}t \\ y=a+\frac{\sqrt{3}}{2}t \end{cases}$ (where $t$ is the parameter and $a\in\mathbb{R}$). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The curve $C$ has the polar equation $\rho=4\cos\theta$. The ray $\theta=\frac{\pi}{3}\;(\rho\geq0)$ intersects curve $C$ at points $O$ and $P$, and line $l$ intersects curve $C$ at points $A$ and $B$.

1. Find the standard equation of line $l$ and the rectangular equation of curve $C$.
2. Find the value of $a$ when $|AB|=|OP|$.

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