
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 line $x = t$ intersects the graphs of functions $f(x) = \ln(x)$ and $g(x) = a + ax - x^2$ at points M and N, respectively. Find the range of real number a such that the difference in the slopes of line OM and ON, $k_{OM} - k_{ON}$, increases monotonically over the interval $t \in [1, +\infty)$.

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