
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
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $$2\sin^{2} \frac {A+B}{2}+\cos2C=1$$  
(1) Find the magnitude of angle $C$;  
(2) If vector $$\overrightarrow {m}=(3a，b)$$ and vector $$\overrightarrow {n}=(a，- \frac {b}{3})$$, with $$\overrightarrow {m} \perp \overrightarrow {n}$$ and $$( \overrightarrow {m}+ \overrightarrow {n})(- \overrightarrow {m}+ \overrightarrow {n})=-16$$, find the values of $a$, $b$, and $c$.

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