
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 quadratic function $f\left(x\right)=ax^{2}+bx+c$ passes through the point $\left(0,3\right)$, and the solution set of the inequality $ax^{2}+bx+c\leqslant 0$ is $\{x\left|\right.1\leqslant x\leqslant 3\}$.  
$(1)$ Find the analytical expression of $f\left(x\right)$;  
$(2)$ If $g\left(x\right)=f\left(x\right)-\left(2t-4\right)x$ has a minimum value of $2$ on the interval $\left[-1,2\right]$, find the value of the real number $t$;  
$(3)$ For any $x\in \left[4,+\infty \right)$, the inequality $\lambda f\left(x\right)-\left(x-2\right) \gt 0$ always holds. Find the range of the real number $\lambda$.

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