
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
Given the function $f(x)=\cos^4x+2\sqrt{3}\sin x \cos x - \sin^4x$.  
$(1)$ Find the smallest positive period and the interval of monotonic decrease of $f(x)$;  
$(2)$ In triangle $ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, if $f(A)=1$ and the median $AD$ of side $BC$ has a length of $\sqrt{7}$, find the maximum area of triangle $ABC$.

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