
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 $$\overrightarrow {m} = (\sqrt {3}\sin\omega x, 1+\cos\omega x)$$, $$\overrightarrow {n} = (\cos\omega x, 1-\cos\omega x)$$, and $f(x) = \overrightarrow {m} \cdot \overrightarrow {n}$, where $\omega > 0$. If the distance from one of the symmetry axes of $f(x)$ to the nearest center of symmetry is $\frac {\pi}{4}$,  
(1) find the center of symmetry of $f(x)$;  
(2) if $g(x) = f(x) + m$ has two different zeros in the interval $[0, \frac {\pi}{2}]$, find the range of the real number $m$.

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