
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 line $l$ is $\begin{cases} x=3+t \\ y=1+at \end{cases}$ (where $t$ is the parameter, $a\in\mathbb{R}$), and the parametric equation of curve $C$ is $\begin{cases} x=2+2\cos \alpha \\ y=2\sin \alpha \end{cases}$ (where $\alpha$ is the parameter). Suppose line $l$ intersects curve $C$ at points $A$ and $B$. When the length of chord $|AB|$ is the shortest, the standard equation of line $l$ is \_\_\_\_\_\_.

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