
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 that vectors $\vec{a}$ and $\vec{b}$ are not collinear, $\overrightarrow{AB}=\lambda \vec{a}+\vec{b}$, and $\overrightarrow{AC}=\vec{a}+\mu \vec{b}$ ($\lambda ,\mu \in R$), determine the relationship between $\lambda$ and $\mu$ that definitely holds if points $A$, $B$, and $C$ are collinear.

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