
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 functions $f(x)=e^{x}$ ($e$ is the base of the natural logarithm) and $g(x)=\frac{a}{2}x+b$ ($a,b \in \mathbb{R}$).

(1) If $h(x)=f(x)g(x)$, $b=1-\frac{a}{2}$, and $a=-4$, find the maximum value of $h(x)$ on $[0,1]$.

(2) If $a=4$, find the range of the real number $b$ such that the equation $f(x)=g(x)$ has exactly two distinct real roots in $[0,2]$.

(3) If $b=-\frac{15}{2}$, find the maximum positive integer $a$ ($a \in \mathbb{N}^+$) such that the graph of $f(x)$ is always above the graph of $g(x)$. (Given $2.71 < e < 2.72$)

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