
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 Cartesian coordinate system, suppose the vertices of triangle ABC are A(0, a), B(b, 0), C(c, 0), and point P(0, p) is on segment AO (excluding the endpoints), where a, b, c, p are all non-zero real numbers. The lines BP and CP intersect AC and AB at points E and F, respectively. A student has correctly derived the equation of line OE: $\left(\frac{1}{b} - \frac{1}{c}\right)x + \left(\frac{1}{p} - \frac{1}{a}\right)y = 0$. Please find the equation of line OF: ______.

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