
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
The function  $f: \mathbb{R}\rightarrow \mathbb{R}$  is such that  $f(x+1)=2f(x)$  for  $\forall$   $x\in \mathbb{R}$  and  $f(x)=x(x-1)$  for  $\forall$   $x\in (0,1]$ . Find the greatest real number  $m$ , for which the inequality  $f(x)\geq -\frac{8}{9}$  is true for  $\forall$   $x\in (-\infty , m]$ .


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