
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
Triangles \(ADL\) and \(BCL\) are similar, \(\frac{AD}{BC}=\frac{AL}{BL}=\frac{LD}{LC}\). Let \(u=BL\) and \(v=AL\). Consider triangle \(ABD\). In the first case: \(AB=6\sqrt{13}\), \(AD=6\), \(BD=\sqrt{36 \cdot 13-36}=12\sqrt{3}\). Then \(\frac{6}{12}=\frac{v}{u}=\frac{12\sqrt{3}-u}{18-v}\), \(u=2v\), \(18-v=24\sqrt{3}-4v\), \(v=8\sqrt{3}-6\), \(BL=16\sqrt{3}-12\).

In the second case: \(AB=30\), \(AD=6\), \(BD=\sqrt{900-36}=12\sqrt{6}\). Then \(\frac{6}{24}=\frac{v}{u}=\frac{12\sqrt{6}-u}{18-v}\), \(u=4v\), \(18-v=48\sqrt{6}-16v\), \(15v=48\sqrt{6}-18\), \(BL=\frac{16\sqrt{6}-6}{5}\).

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