
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 the quadrilateral $ABCD$ in the plane, points $B$ and $D$ are on opposite sides of the line $AC$, $AB=3$, $BC=5$, and the four internal angles are denoted by $A$, $B$, $C$, $D$. Given that $\cos B=-\cos D=\frac{3}{5}$. Find:<br/>
$(1)$ The value of $\angle BAC$;<br/>
$(2)$ The maximum value of the sum of the areas of triangles $\triangle ABD$ and $\triangle ACD$.

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