
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 ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) with one focus \(F\), point \(P\) lies on the \(y\)-axis. The line \(PF\) intersects the ellipse at points \(M\) and \(N\). Given \(\overrightarrow{PM} = \lambda_{1} \overrightarrow{MF}\) and \(\overrightarrow{PN} = \lambda_{2} \overrightarrow{NF}\), find the sum of the real numbers \(\lambda_{1} + \lambda_{2}\).

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