
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is known that $a=3$, $b=4$, and the radius of the circumcircle $R=\frac{5}{2}$, with the length of side $c$ being an integer.

$(1)$ Find the sine value of angle $A$ and the length of side $c$;

$(2)$ There exists a point $D$ on side $AB$, and the line segment $CD$ divides $\triangle ABC$ into two parts with an area ratio of $2:1$. Find the length of segment $CD$.

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