
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 ellipse \(\Gamma\), \(A\) is an endpoint of the major axis, \(B\) is an endpoint of the minor axis, and \(F_{1}, F_{2}\) are the two foci. If \(\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=0\), find the value of \(\tan \angle A B F_{1} \cdot \tan \angle A B F_{2}\).

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