
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 \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. Given the following conditions:
$$
\tan C = \frac{\sin A + \sin B}{\cos A + \cos B}, \quad \sin (B - A) = \cos C
$$

1. Find \( \angle A \) and \( \angle C \).
2. If the area of \( \triangle ABC \) is \( S_{\triangle ABC} = 3 + \sqrt{3} \), then find \( a \) and \( c \).

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