
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 polar coordinate system, the polar coordinate equation of the curve $\Gamma$ is $\rho= \frac {4\cos \theta}{\sin ^{2}\theta}$. Establish a rectangular coordinate system with the pole as the origin, the polar axis as the positive semi-axis of $x$, and the unit length unchanged. The lines $l_{1}$ and $l_{2}$ both pass through the point $F(1,0)$, and $l_{1} \perp l_{2}$. The slope angle of line $l_{1}$ is $\alpha$.
(1) Write the rectangular coordinate equation of the curve $\Gamma$; write the parameter equations of $l_{1}$ and $l_{2}$;
(2) Suppose lines $l_{1}$ and $l_{2}$ intersect curve $\Gamma$ at points $A$, $B$ and $C$, $D$ respectively. The midpoints of segments $AB$ and $CD$ are $M$ and $N$ respectively. Find the minimum value of $|MN|$.

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