
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 the hyperbola $C: \frac{x^2}{a^2} - y^2 = 1$ ($a > 0$) intersect with the line $l: x + y = 1$ at two distinct points $A$ and $B$.

(Ⅰ) Find the range of values for the eccentricity $e$ of the hyperbola $C$.

(Ⅱ) Let the intersection of line $l$ with the y-axis be $P$, and $\overrightarrow{PA} = \frac{5}{12} \overrightarrow{PB}$. Find the value of $a$.

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