
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 triangle $ABC$, let $A$, $B$, and $C$ be its three internal angles and $a$, $b$, and $c$ be their respective corresponding sides. Given that angle $C = \frac{π}{3}$, $c=2$. When $\overrightarrow{AC} \cdot \overrightarrow{AB}$ reaches its maximum value, find the value of $\frac{b}{a}$.

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