
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 functions $f(x)=e^x$ and $g(x)=a\sqrt{x}$ ($a\neq 0$), find the range of values for $a$ if there exists a point $P(x_0, y_0)$ on the graph of $y=f(x)$ such that the tangent line to $y=f(x)$ at point $P(x_0, y_0)$ is also tangent to the graph of $y=g(x)$.

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