
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 $f(x)=x^{3}-ax^{2}$ where $a\in\mathbb{R}$.

(1) If $f'(1)=3$.

$\quad\quad$ (i) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$.

$\quad\quad$ (ii) Find the maximum value of $f(x)$ on the interval $[0,2]$.

(2) If $f(x)+x\geqslant 0$ holds true for all $x\in[0,2]$, find the range of values for the real number $a$.

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