
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
Viral infection refers to the process in which viruses invade the body through various pathways and proliferate in susceptible host cells. If a host is infected with a virus and is treated symptomatically as soon as adverse reactions appear, the quantity of the virus after $x$ hours of medication is given by $f(x)=\left\{\begin{array}{l}-{x}^{2}+10x+5,0≤x≤5,\\-2x+40,5＜x≤20.\end{array}\right.($unit: hundreds of bacteria)<br/>$(1)$ Find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(1,f\left(1\right)\right)$;<br/>$(2)$ Find the time interval during which the quantity of bacteria exceeds $14$ (hundreds).

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