
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 the function $$f(x)=\cos^2 x+ \frac {1}{2}\sin(2x+ \frac {\pi}{2})- \frac {1}{2}$$.
(1) Find the range of values of $f(x)$ on the interval $\left( \frac {\pi}{6}, \frac {2\pi}{3} \right)$.
(2) Let $A$, $B$, and $C$ be the angles of triangle $\triangle ABC$, if angle $C$ satisfies $$f\left( \frac {C}{2}\right)= \frac {\sqrt {2}}{2}$$ and side $c = \sqrt {2}a$, find angle $A$.

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