
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, let the vertices of triangle $ABC$ be $A(0,a)$, $B(b,0)$, $C(c,0)$, and point $P(0,p)$ lies on segment $AO$ (other than the endpoints), where $a$, $b$, $c$, and $p$ are all non-zero real numbers. Lines $BP$ and $CP$ intersect $AC$ and $AB$ at points $E$ and $F$, respectively. A student has correctly calculated the equation of $OE$ as: $\left(\frac{1}{b}-\frac{1}{c}\right)x+\left(\frac{1}{p}-\frac{1}{a}\right)y=0$. Find the equation of $OF$: ______.

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