
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 $ABC$, where $A$, $B$, $C$ are the internal angles and $a$, $b$, $c$ are their respective opposite sides. It is known that $\cos A= \frac{2\sqrt{5}}{5}$ and $\sin B=\frac{\sqrt{10}}{10}$.

1. Find angle $C$.
2. Given $a=\sqrt{10}$, find the area of triangle $ABC$.

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