
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 function $f(x) = -x\ln x + ax$ is increasing on the interval $(0,e)$, where $e$ is the base of natural logarithms. For the function $g(x) = |e^x - a| + \frac{a^2}{2}$ where $x \in [0,\ln 3]$, the difference between the maximum value $M$ and the minimum value $m$ of $g(x)$ is $\frac{3}{2}$. Find the real number $a$.

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