
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 the function $f(x)=a\cdot e^{x-1}$ (where $a$ is a constant), and $f(-1)= \frac {2}{e^{2}}$  
$(1)$ Find the value of $a$;  
$(2)$ Let $g(x)=\begin{cases}f(x), & x < 2 \\ \log_{3}(x-1), & x\geqslant 2\end{cases}$, find the solution set of the inequality $g(x) < 2$.

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