
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 $f(x)=x^{2}-2ax(a > 0)$.  
$(1)$ When $a=2$, solve the inequality $-3 < f(x) < 5$ with respect to $x$;  
$(2)$ For a given positive number $a$, there exists a maximum positive number $M(a)$, such that the inequality $|f(x)|\leqslant 5$ always holds in the interval $[0,M(a)]$. Find the expression for $M(a)$;  
$(3)$ The function $y=f(x)$ has a maximum value of $0$ and a minimum value of $-4$ in the interval $[t,t+2]$. Find the values of the real numbers $a$ and $t$.

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