
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 the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $a\sin B = b\cos\left(A-\frac{\pi}{6}\)$ and $b\cos C = c\cos B$. Find:<br/>
$(1)$ The value of angle $A$;<br/>
$(2)$ If point $D$ lies on side $BC$ and $\cos\angle BAD = \frac{4}{5}$, find the ratio of the areas of triangles $\triangle ABD$ and $\triangle ACD$.

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