
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 an exponential sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, where $a_{n} \gt 0$ and $a_{1}a_{3}=36$, $a_{3}+a_{4}=9(a_{1}+a_{2})$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If ${S_n}+1={3^{{b_n}}}$, find the sequences $\{b_{n}\}$ and $\{a_{n}b_{n}\}$, and the sum of the first $n$ terms $T_{n}$.<br/>$(3)$ Let ${c_n}=\frac{{{a_n}}}{{({{a_n}+1})({{a_{n+1}}+1})}}$, find the sum of the first $2n$ terms of $\{c_{n}\}$, denoted as $P_{2n}$.

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