
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
From point $A$, two rays emerge at an angle $\alpha$. Two points $B$ and $B_{1}$ are taken on one ray, and two points $C$ and $C_{1}$ are taken on the other ray. Find the length of the common chord of the circles circumscribed around triangles $ABC$ and $AB_{1}C_{1}$, if $|AB|-|AC|=\left|AB_{1}\right|-\left|AC_{1}\right|=a$.

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