
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 function $f(x) = \frac{x^2 + a}{x + 1}$ where $a \in \mathbb{R}$:  
(Ⅰ) If the tangent to $f(x)$ at the point $(1, f(1))$ has a slope of $\frac{1}{2}$, find the value of the real number $a$;  
(Ⅱ) If $f(x)$ has an extremum at $x=1$, determine the intervals of monotonicity for the function $f(x)$.

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