
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 the parabola $C: y^2 = 2px$ ($p > 0$) passes through point $A(1, m)$, and $B$ is the intersection point of the directrix of the parabola with the x-axis. If $|AB| = 2\sqrt{2}$,  
(1) Find the equation of the parabola;  
(2) Take any point $P(x_0, 2)$ on the parabola. Draw two lines through $P$ that intersect the parabola at points $M$ and $N$ respectively. Connect $MN$. If the slopes of lines $PM$, $PN$, and $MN$ all exist and are non-zero, and their slopes are $k_1$, $k_2$, and $k_3$ respectively, find the value of $\frac{1}{k_1} + \frac{1}{k_2} - \frac{1}{k_3}$.

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