
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. Find the imaginary part of the complex number $2+\frac{1}{3i}$.
2. Given that $\cos{\frac{π}{3}}=\frac{1}{2}$, $\cos{\frac{π}{5}}\cos{\frac{2π}{5}}=\frac{1}{4}$, $\cos{\frac{π}{7}}\cos{\frac{2π}{7}}\cos{\frac{3π}{7}}=\frac{1}{8}$, ..., find a general expression for the $n^{th}$ equation based on the given equations.
3. If the function $f(x)=\frac{x-2}{e^x}$ reaches an extreme value at $x=x\_0$, find $x\_0$.
4. Given that the function $f(x)$ is an odd function defined on $R$, $f(1)=0$, and $f'(x)>0 (x>0)$, find the solution set of the inequality $f(x)>0$.

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