
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 $xOy$, given point $Q(1, 2)$, $P$ is a moving point, and the slopes of the lines on which the three sides of $\triangle POQ$ lie satisfy $\frac {1}{k_{op}} + \frac {1}{k_{OQ}} = \frac {1}{k_{PQ}}$.  
(1) Find the equation of the trajectory $C$ of point $P$;  
(2) Draw a line $L$ through point $F(1, 0)$ with an inclination angle of $60^\circ$, intersecting curve $C$ at points $A$ and $B$, and find the area of $\triangle AOB$.

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