
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
Consider the function $f(x)=\frac{ln(x+1)}{lnx}$ for $x>1$. Given $f'(x)=\frac{\frac{lnx}{x+1}-\frac{ln(x+1)}{x}}{{(lnx)}^{2}}=\frac{xlnx-(x+1)ln(x+1)}{x(x+1){(lnx)}^{2}}<0$, thus $f\left(x\right)$ is monotonically decreasing on the interval $(1,+\infty)$. Therefore, for $m>n>1$, we have $\frac{ln(m+1)}{lnm}<\frac{ln(n+1)}{lnn}⇒\frac{ln(m+1)}{ln(n+1)}<\frac{lnm}{lnn}⇒log_{n+1}(m+1)<log_{n}m$. Comparing $\log _{2}5$, $\log _{17}626$, and $\frac{5}{2}$ from smallest to largest, the correct order is ______.

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