
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 vectors $\overrightarrow{m}=( \sqrt {3}\sin \frac {x}{4},-1)$ and $\overrightarrow{n}=(\cos \frac {x}{4},\cos ^{2} \frac {x}{4})$, let $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}$.
1. Find the monotonically decreasing interval and the center of symmetry for $f(x)$.
2. In triangle $ABC$, the angles $A$, $B$, and $C$ have opposite sides $a$, $b$, and $c$, respectively. If $f(A)=- \frac {1}{2}$ and $a=2$, find the maximum area of triangle $ABC$.

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