
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 the function $f(x)= \frac {1}{x+2}$, point $O$ is the origin, point $A_{n}(n,f(n))$ where $n \in N^{*}$, vector $ \overrightarrow{i}=(0,1)$, $\theta_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{i}$. Find the range of real number $t$ such that $\frac {\cos \theta_{1}}{\sin \theta_{1}}+ \frac {\cos \theta_{2}}{\sin \theta_{2}}+ \frac {\cos \theta_{3}}{\sin \theta_{3}}+…+ \frac {\cos \theta_{n}}{\sin \theta_{n}} < t$ always holds.

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