
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 $a > 0$, $b > 0$, $\dfrac {\lambda}{a+b}\leqslant \dfrac {1}{a}+ \dfrac {2}{b} (*)$, where $\lambda$ is a constant.  
$(1)$ When $\lambda=6$, is there any $a$, $b$ that makes the inequality $(*)$ not hold? Please explain your reasoning;  
$(2)$ If the inequality $(*)$ always holds for any positive real numbers $a$, $b$, find the maximum value of $\lambda$.

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