
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.

1. If $2\sin A\cos C=\sin B$, find the value of $\frac{a}{c}$.
2. If $\sin (2A+B)=3\sin B$, find the value of $\frac{\tan A}{\tan C}$.

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