
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 that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, and it satisfies the equation $S\_{n-1} + S\_n = 2n^2 + 1$ for $n \geq 2, n \in \mathbb{N}^+$, and $a\_1 = x$, and $\{a\_n\}$ is monotonically increasing, determine the range of values for $x$.

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