
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
In the triangular prism \(ABC-A_{1}B_{1}C_{1}\), the square \(AA_{1}C_{1}C\) has a side length of 4. The plane \(ABC\) is perpendicular to the plane \(AA_{1}C_{1}C\), \(AB = 3\), and \(BC = 5\). If there is a point \(D\) on the line segment \(BC_{1}\) such that \(AD \perp A_{1}B\), find the value of \(\frac{BD}{BC_{1}}\).

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