
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 $f(x)=e^{x}$ and $g(x)=ax+b$, where $a,b \in \mathbb{R}$.
(1) Let $h(x)=xg(x)+1$.
$\quad\quad$ (a) If $a \neq 0$, under what conditions for $a$ and $b$ do the curves $y=f(x)$ and $y=h(x)$ always have the same tangent line at $x=0$?
$\quad\quad$ (b) When $a=1$, find the monotonic intervals of the function $F(x)=\frac{h(x)}{f(x)}$.
(2) If the set $\{x|f(x) < g(x)\}$ is empty, find the maximum value of $ab$.

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