
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 following conditions to supplement the question below and solve it:

In triangle ABC, where the opposite sides of the internal angles A, B, and C are a, b, and c, respectively.

$(1)$ 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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