
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 functions $f(x)=e^{x}-x$ and $g(x)=e^{-x}+x$, where $e$ is the base of the natural logarithm.  
$(1)$ Find the equation of the tangent line to the graph of the function $y=f(x)$ at the point $(1,f(1))$.  
$(2)$ Let $F(x)=af(x)-g(x)$.  
① If $a=e$, find the intervals where the function $y=F(x)$ is monotonic, and write the range of real numbers $m$ for which the function $y=F(x)-m$ has three zeros.  
② When $0 < a < 1$, let $x_{1}$ and $x_{2}$ be the points of maximum and minimum values of the function $y=F(x)$, respectively. If the inequality $F(x_{1})+tF(x_{2}) > 0$ holds for any $a \in (0,1)$, find the range of real numbers $t$.

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