
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
Consider a set of points $(x,y)$ parameterized by $t$, taking all real values, defined by the following equations:
\[
x = t^2 + k
\]
\[
y = t^3 - kt + 5
\]
Determine the value of $k$ for which the curve crosses itself and calculate the corresponding $(x,y)$ pair where this crossing occurs.

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