
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 two lines $l\_1$: $y=m$ and $l\_2$: $y= \frac {8}{2m+1}(m > 0)$, line $l\_1$ intersects with the graph of the function $y=|\log _{2}x|$ from left to right at points $A$, $B$, and line $l\_2$ intersects with the graph of the function $y=|\log _{2}x|$ from left to right at points $C$, $D$. Let $a$ and $b$ be the lengths of the projections of line segments $AC$ and $BD$ on the $X$-axis, respectively. Find the minimum value of $\frac {b}{a}$ as $m$ varies.

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