
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 $f(x)=e^{x}+2e^{-x}+a$ and $g(x)=x^{2}-x+1$ have a common tangent line $y=mx+1$.

$(1)$ Find the values of the real numbers $m$ and $a$.

$(2)$ If ${e^{{x_1}}}+{e^{{x_2}}}=3$, and $f(x_{1})\cdot f(x_{2})\geqslant 3(x_{1}+x_{2}+k)$, find the maximum value of the real number $k$.

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