
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 $\triangle ABC$, with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, vectors $\overrightarrow {m}=(2\cos \frac {C}{2}, -\sin C)$, $\overrightarrow {n}=(\cos \frac {C}{2}, 2\sin C)$, and $\overrightarrow {m} \perp \overrightarrow {n}$.
(1) Find the magnitude of angle $C$;
(2) If $a^2 =2b^2+c^2$, find the value of $\tan A$.

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