
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 $\overset{→}{a} =( \sin x, -1)$, $\overset{→}{b} =( \sqrt{3}\cos x, -\frac{1}{2})$, and the function $f(x)=( \overset{→}{a} + \overset{→}{b} )⋅ \overset{→}{a} -2$.

1. Find the monotonically decreasing interval of the function $f(x)$ for $x\in[0,π]$.
2. Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively, of triangle $ABC$, where $A$ is an acute angle, $a=2 \sqrt{3}$, $c=4$, and $f(A)=1$, find the area of triangle $ABC$.

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