
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 function $f(x)=\left\{\begin{array}{l}{-{x}^{2}+2ax-{a}^{2}+2a, x>0}\\{{x}^{2}+2ax+{a}^{2}-2a, x<0}\end{array}\right.$ $(a > 0)$.<br/>$(1)$ When $a=2$, find the value of $f(f(5))$;<br/>$(2)$ If the graph of the function $f(x)$ intersects the line $y=3$ at three distinct points, write down the range of real numbers for $a$ directly;<br/>$(3)$ Under the conditions of $(2)$, let the abscissas of the three intersection points be $x_{1}$, $x_{2}$, $x_{3}$ ($x_{1} < x_{2} < x_{3}$). If $\frac{x_1}{x_2+x_3} < t$ always holds, find the range of real number $t$.

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