
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
1. Calculate the value of the expression $\left(\frac{\sqrt{2}}{1-i}\right)^{2018} + \left(\frac{1+i}{1-i}\right)^{6}$.
2. If $f(x)=x^{3}+ax^{2}+bx+a^{2}$ has a local extremum of $10$ at $x=1$, find the value of $a+b$.
3. Find the equation of the tangent line to the curve $f(x)=\ln{x}-\frac{2^{x}}{\ln{2}}$ at $x=1$.
4. Given that $f'''(x)$ is the derivative of $f(x)$ and $f'''(x)=e^{x}(2x+3)+f(x)$ for all real numbers $x$, where $e$ is the base of the natural logarithm, and $f(0)=1$. If the solution set of the inequality $f(x)-k<0$ contains exactly two integers, find the range of $k$.

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