
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 point \(C\) on the line segment \(AB\) and point \(P\) outside the line segment \(AB\), such that \(|\overrightarrow{PA}| - |\overrightarrow{PB}| = 2\), \(|\overrightarrow{PA} - \overrightarrow{PB}| = 2\sqrt{5}\), and \(\frac{\overrightarrow{PA} \cdot \overrightarrow{PC}}{|\overrightarrow{PA}|} = \frac{\overrightarrow{PB} \cdot \overrightarrow{PC}}{|\overrightarrow{PB}|}\), let \(I\) be a point on \(PC\) such that \(\overrightarrow{BI} = \overrightarrow{BA} + \lambda\left(\frac{\overrightarrow{AC}}{|\overrightarrow{AC}|} + \frac{\overrightarrow{AP}}{|\overrightarrow{AP}|}\right)\) with \(\lambda > 0\). Determine the value of \(\frac{\overrightarrow{BI} \cdot \overrightarrow{BA}}{|\overrightarrow{BA}|}\).

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