
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
If the function $y=f\left(x\right)$ is an odd function on $R$, and the function $F\left(x\right)=af\left(x\right)+bx+1$ has a maximum value of $2$ on $\left(0,+\infty \right)$, determine the behavior of the function $y=F\left(x\right)$ on $\left(-\infty ,0\right)$.

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