
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
Choose one of the two conditions in (1) $2a-b=2c\cos B$, (2) $\sqrt{3}c\cos A+asinC=\sqrt{3}b$, and complete the following question, then find the answers to (1) and (2).

Question: In triangle $\triangle ABC$, where the sides opposite the three interior angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, it is known that ______

(1) Find angle $C$;

(2) If point $D$ satisfies $2AD=DB$ and $CD=1$, find the maximum area of triangle $\triangle ABC$.

(Note: If you choose to answer using both conditions separately, the scoring will be based on the first answer.)

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