
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 $\overrightarrow{a} = (2\cos x, \sin x)$ and $\overrightarrow{b} = (\cos x, 2\sqrt{3}\cos x)$, define the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - 1$.
- (I) Determine the interval where the function $f(x)$ is monotonically decreasing.
- (II) In an acute triangle $\triangle ABC$, with the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively, where $\tan B = \frac{\sqrt{3}ac}{a^{2}+c^{2}-b^{2}}$, for any angle $A$ that satisfies the condition, find the range of $f(A)$.

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