
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 curve $C_{1}: y=e^{x}$ has a point $A(x_{1},y_{1})$, and curve $C_{2}: y=1+\ln(x-a) (a > 0)$ has a point $B(x_{2},y_{2})$. When $y_{1}=y_{2}$, for any $x_{1}$, $x_{2}$, it always holds that $|AB| \geqslant e$. Find the minimum value of $a$.

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