
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 the angles $A, B, C$ are $a, b, c$ respectively. Derive the projection theorem: $c = a \cdot \cos B + b \cdot \cos A$; the sine rule: $\frac{a}{\sin A}=\frac{b}{\sin B}$; the cosine rule: $c^{2} = a^{2} + b^{2} - 2ab \cdot \cos C$.

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