
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
If \( l \) and \( l' \) are the bisectors of the angle \( C \) of a triangle and its exterior angle, respectively, then

\[ 
l^{\prime 2} + l^{2} = \frac{64 R^{2} S^{2}}{\left(a^{2} - b^{2}\right)^{2}} 
\]

where \( R \) is the radius of the circumcircle of the triangle and \( S \) is the area of the triangle.

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