
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 $\triangle ABC$, $A=30^{\circ}$, $BC=2\sqrt {5}$, point $D$ lies on side $AB$, and $\angle BCD$ is acute. Also, $CD=2$, and the area of $\triangle BCD$ is $4$.
1. Find the value of $\cos \angle BCD$.
2. Find the length of side $AC$.

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