
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 acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that angle $A = \frac{π}{3}$. If $AM$ is the angle bisector of $\angle CAB$ and intersects $BC$ at $M$, and $AM = 2$, then the minimum value of $AC + 3AB$ is ______; if the circumcenter of $\triangle ABC$ is $O$ with a radius of $1$, then the range of $\overrightarrow{OA} \cdot (\overrightarrow{AB} + \overrightarrow{AC})$ is ______.

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