
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
Let $f(x) = x^2\ln{x}$. By the product rule for differentiation, $(x^2\ln{x})' = 2x\ln{x} + x$. Integrating both sides over the interval $[1, e]$, we have: $$\int_{1}^{e} (x^2\ln{x})'dx = \int_{1}^{e} 2x\ln{x}dx + \int_{1}^{e} xdx.$$  
Rearranging, we get: $$\int_{1}^{e} 2x\ln{x}dx = (x^2\ln{x}) \bigg|_{1}^{e} - \int_{1}^{e} xdx = e^2 - \left( \frac{1}{2}e^2 - \frac{1}{2} \right) = \frac{1}{2}e^2 + \frac{1}{2}.$$  
This method of finding definite integrals is called integration by parts. Please calculate the following definite integral using the method above: $$\int_{1}^{e} \ln{x}dx = \boxed{\text{\_\_\_\_\_\_}}.$$

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