
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 $\overrightarrow {m}=( \sqrt {3}\cos x, 1)$ and $\overrightarrow {n}=(\sin x, \cos^{2} x - 1)$, and a function $f(x) = \overrightarrow {m} \cdot \overrightarrow {n} + \frac{1}{2}$,
(1) If $x \in [0, \frac{\pi}{4}], f(x) = \frac{\sqrt{3}}{3}$, find the value of $\cos 2x$;
(2) In $\triangle ABC$, angles $A$, $B$, $C$ are opposite to sides $a$, $b$, $c$ respectively, and they satisfy $2b \cos A \leq 2c - \sqrt{3}a$. When $B$ is at its maximum value, $a = 1$, and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{4}$, find the value of $\frac{a + c}{\sin A + \sin C}$.

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