
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 $f(x)=ax^2-e^x$ $(a \in \mathbb{R})$.
1. If the tangent to the curve $y=f(x)$ at $x=1$ is perpendicular to the $y$-axis, find the maximum value of $y=f'(x)$;
2. If for any $0 \leqslant x_1 < x_2$, it holds that $f(x_2) + x_2(2-2\ln 2) < f(x_1) + x_1(2-2\ln 2)$, find the range of $a$.

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