
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 $x$ such that the matrix $\begin{pmatrix}x^2 & 2x+1\\ 3x & 4x+2\end{pmatrix}$ has a determinant equal to 10.
**A)** $x = 2$  
**B)** $x = 2$ and $x = -1$  
**C)** $x = 2$, $x = -1$, and $x = 1$  
**D)** $x = -2$  
**E)** None of these

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