
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 $f\left( x \right)={x}^{2}+3x+a$ and $g\left( x \right)=\frac{f\left( x \right)}{x+1}$,

(1) If the solution set of the inequality $f\left( x \right) < 0$ is $\left\{ x\left| a < x < 1 \right. \right\}$, find the value of $a$;

(2) When $g(b)=b+ \frac{1}{2}a $, where $ab < 0$, find the maximum value of $4a+b$;

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