
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 proposition P: The function $$f(x)= \frac {1}{3}(1-x)$$ and $|f(a)|<2$, proposition Q: Set $A=\{x|x^2+(a+2)x+1=0, x\in \mathbb{R}\}$, $B=\{x|x>0\}$ and $A\cap B=\emptyset$,

(1) Find the range of real numbers $a$ for which propositions P and Q are true, respectively;  
(2) For what range of real numbers $a$ is exactly one of the propositions P and Q true;  
(3) Let the range of $a$ for which both P and Q are true be the set $S$, $$T=\{y|y=x+ \frac {m}{x}, x\in \mathbb{R}, x\neq 0, m>0\}$$, if $\complement_{\mathbb{R}}T \subseteq S$, find the range of $m$.

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