
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 $\overrightarrow{m}=(\sin \omega x+\cos \omega x, \sqrt {3}\cos \omega x)$ and $\overrightarrow{n}=(\cos \omega x-\sin \omega x,2\sin \omega x)(\omega > 0)$, the function is defined as $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}$. The distance between adjacent symmetry axes of $f(x)$ is no less than $\frac {\pi}{2}$.

1. Find the range of $\omega$.
2. In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, with $a=2$. When $\omega$ is at its maximum value and $f(A)=1$, find the maximum area of $\triangle ABC$.

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