
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 $a$, $b$, and $c$ respectively, and it is given that $1 + \frac{\tan A}{\tan B} = \frac{2c}{b}$.
(1) Find angle $A$.
(2) Given vectors $\overrightarrow{m} = (0, -1)$ and $\overrightarrow{n} = (\cos B, 2\cos^2 \frac{C}{2})$, find the minimum value of $|\overrightarrow{m} + \overrightarrow{n}|$.

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