
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 Cartesian coordinate system $xOy$, given points $A(-2, 0)$, $B(0, 2\sqrt{3})$, and $C(2\cos\theta, 2\sin\theta)$, where $\theta \in [0, \frac{\pi}{2}]$.  
Let point $D(1, 0)$. Find the maximum value of $\frac{\overrightarrow{AC}}{\overrightarrow{BD}}$.  
Let point $E(a, 0)$, where $a \in \mathbb{R}$. Express $\overrightarrow{OC} \cdot \overrightarrow{CE}$ as a function of $\theta$, and denote its minimum value as $f$. Find the expression for $f$ and its maximum value.

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