
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 that \( A \) and \( B \) are the common vertices of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a > b > 0)\) and the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}=1 \ (a > 0, b > 0)\), let \( P \) and \( M \) be points on the hyperbola and ellipse respectively, different from \( A \) and \( B \), such that \(\overrightarrow{AP} + \overrightarrow{BP} = \lambda(\overrightarrow{AM} + \overrightarrow{BM})\), where \(\lambda \in \mathbb{R}, |\lambda| > 1\). 

If the slopes of the lines \( AP \), \( BP \), \( AM \), \( BM \) are \( k_{1}, k_{2}, k_{3}, k_{4} \) respectively, and \( k_{1} + k_{2} = 5 \), find \( k_{3} + k_{4} \).

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