
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 conditions from the following: $b-ccosA=a(\sqrt{3}sinC-1)$; $sin(A+B)cos(C-\frac{π}{6})=\frac{3}{4}$, and supplement it in the question below, then solve it. In triangle $\triangle ABC$: the sides opposite to the internal angles $A$, $B$, $C$ are $a$, $b$, $c$, ___.  
$(1)$ Find the measure of angle $C$;  
$(2)$ Let $D$ be the midpoint of side $AB$, find the maximum value of $\frac{CD^2}{a^2+b^2}$.

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