
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 plane rectangular coordinate system \(xOy\), \(F\) is the focus of the parabola \(\Gamma: y^2 = 2px (p > 0)\). Point \(B\) is on the \(x\)-axis, and is to the right of point \(F\). Point \(A\) is on \(\Gamma\), and \(|AF| = |BF|\). The second intersection points of the line \(AF\) and the line \(AB\) with \(\Gamma\) are \(M\) and \(N\) respectively. If \(\angle AMN = 90^\circ\), find the slope of the line \(AF\).

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