
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 $$f(x)= \frac {1}{2}\cos(2x-\varphi)  (0<\varphi<\pi)$$, its graph passes through the point $$( \frac {\pi}{6}, \frac {1}{2})$$.  
(1) Find the value of $\varphi$;  
(2) If the graph of the function $y=f(x)$ is transformed by reducing the x-coordinates of all points to half of their original values while keeping the y-coordinates unchanged, resulting in the graph of the function $y=g(x)$, find the range of $y=g(x)$ for $x\in[0, \frac {\pi}{4}]$.

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