
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 curve $C: f(x)=x+ \frac {a}{x}$ ($a>0$), and the line $l: y=x$, there is a moving point $P$ on curve $C$. Perpendiculars are drawn from point $P$ to line $l$ and the y-axis, with the feet of the perpendiculars being $A$ and $B$, respectively. A tangent line to curve $C$ at point $P$ intersects line $l$ and the y-axis at points $M$ and $N$, respectively, with $O$ being the origin. If the area of $\triangle ABP$ is $\frac {1}{2}$, then the area of $\triangle OMN$ is.

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