
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 plane, let the vertices of triangle $ABC$ be $A(0,a)$, $B(b,0)$, $C(c,0)$, and point $P(0,p)$ be on line segment $AO$ (and not an endpoint). Given that $a$, $b$, $c$, and $p$ are non-zero real numbers, lines $BP$ and $CP$ intersect $AC$ and $AB$ at points $E$ and $F$, respectively. One student has correctly derived the equation of $OE$ as $(\frac{1}{b} - \frac{1}{c})x + (\frac{1}{p} - \frac{1}{a})y = 0$. Find the equation of $OF$: $(\_\_\_\_\_\_\_)x + (\frac{1}{p} - \frac{1}{a})y = 0$.

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