
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 hyperbola $C$: $\frac{x^2}{1-\lambda} - \frac{y^2}{\lambda} = 1 (0 < \lambda < 1)$ with its right focus $B$ and $O$ as the coordinate origin.
1) If a line perpendicular to the $x$-axis passes through point $B$ and intersects the hyperbola $C$ at points $M$ and $N$, such that $|MN| = 2$, find the value of $\lambda$.
2) When $\lambda = \frac{3}{4}$, a line $l$ with a slope angle of $\frac{\pi}{3}$ passes through point $B$ and intersects the hyperbola $C$ at point $A$. Find the length of $OA$.

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