
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 plane vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\vec{a}| = |\vec{b}|$, $\vec{c} = \lambda \vec{a} + \mu \vec{b}$, $|\vec{c}| = 1 + \vec{a} \cdot \vec{b}$, and $(\vec{a} + \vec{b}) \cdot \vec{c} = 1$, find the minimum possible value of $\frac{|\vec{a} - \vec{c}|}{|1 + \mu|}$.

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