
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given that $a+b=5$ and $\left(2a+b\right)\cdot \cos C+c\cdot \cos B=0$.

(1) If the area of $\triangle ABC$ is $\dfrac{\sqrt{3}}{2}$, find $c$.

(2) If point $D$ is the midpoint of segment $AB$ and $\angle ACD=30^{\circ}$, find $a$ and $b$.

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