
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 line $l: x+y=1$ and the hyperbola $C: \frac{x^2}{a^2} - y^2 = 1$ ($a > 0$).

(1) If $a = \frac{1}{2}$, find the length of the chord formed by the intersection of $l$ and $C$.

(2) If $l$ and $C$ have two distinct intersection points, find the range of values for the eccentricity $e$ of the hyperbola $C$.

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