
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 equation $x^{2}-(k+2)x+2k=0$.  
$(1)$ Please determine the solutions of the equation.  
$(2)$ If the sides of an isosceles triangle $ABC$ are $a=1$, and the other two sides $b$ and $c$ are exactly the two roots of this equation, find the perimeter of $\triangle ABC$.

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