
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 $f(x)=x^{2}-2x-3$.  
$(1)$ Let $g(x)=f(x)+a$. If the graph of the function $g(x)$ does not intersect the $x$-axis, find the range of real numbers for $a$.  
$(2)$ Let $h(x)=x+\frac{25}{x}+b$. If for any $x_{1}\in [1,4]$, there exists $x_{2}\in (1,5]$ such that $f(x_{1})=h(x_{2})$, find the range of real numbers for $b$.

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