
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 \( a \in \mathbf{R} \) and \( b \in \mathbf{R}_{+} \), where \( \mathrm{e} \) is the base of the natural logarithm, find the minimum value of \( \left( \mathrm{e}^{a} - \ln b \right)^{2} + (a - b)^{2} \).

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