
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
If the graph of $$f(x)=\sin(2\omega x- \frac {\pi}{6})$$ is symmetric about the line $$x= \frac {\pi}{3}$$, where $$\omega\in(- \frac {1}{2}, \frac {5}{2})$$,  
(1) Find the expression for $f(x)$;  
(2) Move the graph of $y=f(x)$ to the left by $$\frac {\pi}{3}$$ units, and then double the x-coordinates of the resulting graph (keeping the y-coordinates unchanged) to get the graph of $y=g(x)$. Find the expression for $y=g(x)$;  
(3) If the graph of the function $y=g(x)$ ($$x\in( \frac {\pi}{2},3\pi)$$) intersects the graph of $y=a$ at three points and the x-coordinates of the intersection points form a geometric sequence, find the value of $a$.

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