
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
If $P$ is a point on the left branch of the hyperbola $x^2 - y^2 = \lambda$ ($\lambda > 0$), and $F_1$, $F_2$ are the left and right foci respectively, given that $|PF_2| = 6$ and $PF_1$ is perpendicular to the real axis of the hyperbola, find the value of $\lambda$.

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