
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 triangle \( \triangle ABC \) with internal angles \( A, B, C \) opposite to sides \( a, b, c \) respectively, vectors \( \vec{m} = (1, 1 - \sqrt{3} \sin A) \) and \( \vec{n} = (\cos A, 1) \), and \( \vec{m} \perp \vec{n} \):
(1) Find angle \( A \);
(2) Given \( b + c = \sqrt{3} a \), find the value of \( \sin \left(B + \frac{\pi}{6} \right) \).

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