
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
Given the polar coordinate system and the Cartesian coordinate system $xOy$ use the same unit of length, and the origin $O$ is the pole, with the $x$-axis as the positive semi-axis. The Cartesian equation of curve $C_{1}$ is $\frac{x^{2}}{3}+y^{2}=1$, the Cartesian equation of line $l$ is $x+y-4=0$, and the polar equation of curve $C_{2}$ is $\rho= \frac{1}{1-\cos \theta}$.
(Ⅰ) Find a point $P$ on curve $C_{1}$ such that the distance from point $P$ to line $l$ is maximized;
(Ⅱ) Draw two mutually perpendicular lines through the pole $O$, intersecting curve $C_{2}$ at points $A$, $B$ and $C$, $D$ respectively, and find the minimum value of $|AB|+|CD|$.

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