
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 $AA_1$, $BB_1$, and $CC_1$ are the angle bisectors of triangle $ABC$, and points $A_1$, $B_1$, and $C_1$ lie on sides $BC$, $CA$, and $AB$ respectively. Denote $BC=a$, $CA=b$, and $AB=c$. If points $A_1$, $B_1$, $C_1$, and $B$ are concyclic, find the minimum value of $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$.

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