
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 $f(x)=e^{x}-ax^{2}$, the equation of the tangent line to the curve $y=f(x)$ at $(1,f(1))$ is $y=bx+1$.  
(I) Find the values of $a$ and $b$;  
(II) Find the maximum value of $f(x)$ on $[0,1]$;  
(III) When $x \in \mathbb{R}$, determine the number of intersection points between $y=f(x)$ and $y=bx+1$. (Only the conclusion is required, no proof needed.)

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