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

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