
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 statement below and answer the questions:

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

$(1)$ Find angle C;  
$(2)$ If the area of triangle ABC is $8\sqrt{3}$ and AC's midpoint is D, find the minimum value of BD.

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