
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
(1) Given that $x > 0$ and $y > 0$, if $\sqrt {2}$ is the geometric mean of $2^{x}$ and $4^{y}$, find the minimum value of $x^{2}+2y^{2}$.

(2) Given that $m > 0$, $n > 0$, and $m+n=1$, find the minimum value of $\dfrac {m^{2}}{m+2}+ \dfrac {n^{2}}{n+1}$.

(3) Given that $a+b=2$ and $b > 0$, find the minimum value of $\dfrac {1}{2|a|}+ \dfrac {|a|}{b}$.

(4) Based on the solutions to the above problems, summarize the strategies for solving conditional equations to find the extreme values.

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