
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 function $f(x)= \begin{vmatrix} x & \frac {1}{x} \\ -2 & 1\end{vmatrix}$ ($x>0$), its range is set $A$,  
(1) If the universal set $U=\mathbb{R}$, find $C_UA$;  
(2) For any $x \in (0, \frac {1}{2}]$, the inequality $f(x)+a \geq 0$ always holds, find the range of the real number $a$;  
(3) Let $P$ be any point on the graph of the function $f(x)$, and perpendiculars are drawn from point $P$ to the line $y=x$ and the y-axis, with the feet of the perpendiculars being $A$ and $B$ respectively, find the value of $\overrightarrow {PA} \cdot \overrightarrow {PB}$.

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