
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 $f(x) = \ln x + ax - x^2$ where $0 < a \leq 1$:
(I) When $a = \frac{1}{2}$, find the equation of the tangent line to the graph of $f(x)$ at the point $(1, f(1))$.
(II) Suppose the interval on which the function $f(x)$ is monotonically increasing is $(s, t)$ where $s < t$, find the maximum value of $t-s$.

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