
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 $xOy$, a polar coordinate system is established with the origin $O$ as the pole and the positive half-axis of the x-axis as the polar axis. It is known that the point $P(\sqrt {2}, \frac {7\pi}{4})$ lies on the line $l: \rho\cos\theta +2\rho\sin\theta +a=0$ ($a\in\mathbb{R}$).
(Ⅰ) Find the Cartesian equation of line $l$.
(Ⅱ) If point $A$ lies on the line $l$, and point $B$ lies on the curve $C: \begin{cases} x=t \\ y=\frac{1}{4}t^2 \end{cases}$ (where $t$ is a parameter), find the minimum value of $|AB|$.

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