
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
Let $f(x)$ and $g(x)$ be two functions defined on the same interval $[a, b]$. If for any $x \in [a, b]$, the inequality $|f(x) - g(x)| \leq 1$ holds, then $f(x)$ and $g(x)$ are considered "close functions" on $[a, b]$. Given $f(x) = x^2 + x + 2$ and $g(x) = 2x + 1$, determine the possible "close interval".

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