
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 $f(x)=\ln x- \frac{1}{2}ax^{2}+bx (a > 0), f′(1)=0$.

1. Express $b$ using a formula that involves $a$.
2. Let $F(x)=f(x)+ \frac{1}{2}ax^{2}-bx+ \frac{a}{x} (0 < x\leqslant 3)$. On the graph of this function, the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ is always less than or equal to $\frac{1}{2}$. Determine the range of values for the real number $a$.
3. If $a=2$, find the maximum value of $f(x)$ in the interval $[c,c+ \frac{1}{2}] (c > 0)$.

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