
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 a triangle $ABC$ with interior angles $A, B, C$ and corresponding opposite sides $a, b, c$ respectively, and given that angle $C = \frac{\pi}{3}$ and side $c=2$. Find the value of $\frac{b}{a}$ when the dot product $\overrightarrow{AC} \cdot \overrightarrow{AB}$ is maximum.

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