
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
Determine the values of $k$ for which the equation $3x^2 - (k+1)x + 2 = 0$ will have real and equal roots.
A) $2\sqrt{6} - 1$ only
B) $-2\sqrt{6} - 1$ only
C) $2\sqrt{6} - 1$ and $-2\sqrt{6} - 1$
D) $-\sqrt{6} + 1$ and $\sqrt{6} - 1$

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