
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 $f(x)=x^{2}+2ax+1(a∈R)$, where $f′(x)$ is the derivative of $f(x)$.

(1) Find the range of values for $a$ if the inequality $f(x)≤f′(x)$ always holds true when $x∈[-2,-1]$;

(2) Solve the equation $f(x)=|f′(x)|$;

(3) Define function $g(x)= \begin{cases} f′(x), & \text{if $f(x)≥f′(x)$} \\ f(x), & \text{if $f(x)<f′(x)$} \end{cases}$, find the minimum value of $g(x)$ when $x∈[2,4]$.

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