
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 a quadratic function $f(x)=ax^{2}+bx+c$ whose graph passes through the point $(-2,0)$, and the inequality $2x \leqslant f(x) \leqslant \frac{1}{2}x^{2}+2$ holds for all real numbers $x$  
$(I)$ Find the expression for the function $f(x)$;  
$(II)$ If for any $x \in [-1,1]$, the inequality $f(x+t) < f\left( \frac{x}{3}\right)$ always holds, find the range of the real number $t$.

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