
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
Let $F_1$ and $F_2$ be the left and right foci of the hyperbola $\frac{x^2}{1} - \frac{y^2}{4} = 1$, respectively. If there exists a point $P$ on the right branch of the hyperbola such that $(\vec{OP} + \vec{OF_2}) \cdot \vec{F_2P} = 0$ (where $O$ is the origin), and $|PF_1| = \lambda |PF_2|$, then the value of $\lambda$ is

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