
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 $f(x)=2|x+1|-|x-1|$, its minimum value is $k$.

(1) Find the value of $k$;

(2) If $a$, $b$, $c\in R$, ${{a}^{2}}+{{c}^{2}}+\dfrac{{{b}^{2}}}{2}=-k$, find the maximum value of $b(a+c)$.

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