
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
The lengths of the sides of a triangle are successive terms of a geometric progression. Let \( A \) and \( C \) be the smallest and largest interior angles of the triangle respectively. If the shortest side has length \( 16 \mathrm{~cm} \) and
$$
\frac{\sin A - 2 \sin B + 3 \sin C}{\sin C - 2 \sin B + 3 \sin A} = \frac{19}{9},
$$
find the perimeter of the triangle in centimetres.

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