
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 $f(x)=(2ax^2+bx+1)e^{-x}$ (where $e$ is the base of the natural logarithm).

(Ⅰ) If $a=\frac{1}{2}, b\geqslant 0$, find the monotonic intervals of the function $f(x)$;

(Ⅱ) If $f(1)=1$, and the equation $f(x)=1$ has a solution in $(0,1)$, find the range of the real number $a$.

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