
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
Euler's formula \(e^{ix}=\cos x + i\sin x\) (where \(i\) is the imaginary unit) was developed by the renowned mathematician Euler. He expanded the domain of the exponential function to the complex plane and established the relationship between trigonometric functions and exponential functions. This formula holds a very important status in the theory of complex variables and is heralded as the "mathematical bridge". According to Euler's formula, if the complex number expressed by \(e^{\frac{\pi}{2}i}\) is denoted by \(z\), calculate the value of \(z\cdot(1+2i)\).

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