
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
It is known that the odd terms of the sequence $\{a_n\}$ form an arithmetic sequence, and the even terms form a geometric sequence, with both the common difference and the common ratio being $2$. If for any positive integers $m$, $n$ satisfying $m+n \leqslant 5$, it holds that $a_m + a_n = a_{m+n}$.

$(I)$ Find the general formula for the sequence $\{a_n\}$;  
$(II)$ If $b_n= \begin{cases} \frac {a_n+1}{a_n^2a_{n+2}^2}, & \text{for odd } n \\ \frac {1}{a_n^2}, & \text{for even } n\end{cases}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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