
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 vectors $$\overrightarrow {m}=(2\cos\omega x，-1)$$, $$\overrightarrow {n}=(\sin\omega x-\cos\omega x，2)$$ ($\omega>0$), and the function $f(x)= \overrightarrow {m}\cdot \overrightarrow {n}+3$, if the distance between two adjacent centers of symmetry of the graph of function $f(x)$ is $\frac {\pi}{2}$.  
(Ⅰ) Find the interval of monotonic increase for the function $f(x)$.  
(Ⅱ) If the graph of function $f(x)$ is first translated to the left by $\frac {\pi}{4}$ units, and then the horizontal coordinate is shortened to half of its original length without changing the vertical coordinate, to obtain the graph of function $g(x)$, find the range of $g(x)$ when $x\in[ \frac {\pi}{4}， \frac {\pi}{2}]$.

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