
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 $\overrightarrow {a} = (\sin\omega x + \cos\omega x, \sqrt {3}\cos\omega x)$, $\overrightarrow {b} = (\cos\omega x - \sin\omega x, 2\sin\omega x)$ ($\omega > 0$), if the distance between two adjacent axes of symmetry of the function $f(x) = \overrightarrow {a} \cdot \overrightarrow {b}$ equals $\frac {\pi}{2}$.

(Ⅰ) Find the value of $\omega$;  
(Ⅱ) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $f(C) = 1$, $c = 2$, and $\sin C + \sin (B - A) = 3\sin 2A$, find the area of $\triangle ABC$.

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