
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 $a \in \mathbb{R}$, consider the hyperbola $\Gamma: \frac{x^2}{a^2}-y^2 = 1$.
1. If the point $(2,1)$ lies on $\Gamma$, find the coordinates of the foci of $\Gamma$.
2. If $a = 1$, and the line $y = kx + 1$ intersects $\Gamma$ at points $A$ and $B$, with the midpoint of the segment $AB$ having an $x$-coordinate of $1$, find the value of the real number $k$.

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