
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 the function $f(x)$ defined on $(0, +\infty)$ and satisfying the equation $xf'(x) = 1 + x$ with $f(1) = 2$, find the range of positive real values of $a$ for which the inequality $f(x) \geq (a + 1)x + 1$ has a solution.

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