
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 quadratic equation in terms of $x$: $x^{2}-\left(k+1\right)x+\frac{1}{4}{k^2}+1=0$.  
$(1)$ For what values of $k$ does the equation have two real roots?  
$(2)$ If the two roots of the above quadratic equation are the lengths of two adjacent sides of a rectangle, and the diagonal of this rectangle is $\sqrt{5}$, find the value of $k$.

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