
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 line $l$ with an inclination angle of $45^\circ$ that passes through point A$(1, -2)$ and point B, where point B is in the first quadrant, and $|AB| = 3\sqrt{2}$.

(Ⅰ) Find the coordinates of point B;  
(Ⅱ) If the line $l$ intersects the hyperbola $C: \frac{x^2}{a^2} - y^2 = 1 (a > 0)$ at points E and F, and the midpoint of segment EF has coordinates $(4, 1)$, find the value of $a$.

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