
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 parabola $C: y= \frac {1}{2}x^{2}$, a moving line passing through point $Q(1,1)$ intersects the parabola $C$ at two distinct points $A$ and $B$. Tangent lines to the parabola at points $A$ and $B$ are $l_{1}$ and $l_{2}$, respectively, and these lines intersect at point $P$.

(Ⅰ) Find the equation of the trajectory of point $P$;  
(Ⅱ) Find the minimum value of the area of $\triangle PAB$ and the equation of line $AB$ at this time.

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