
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
Let $a_n$ be a monotonically increasing arithmetic sequence, where $a_1=1$, and $a_3$, $a_6$, $a_{12}$ form a geometric sequence.

(1) Find the general term formula $a_n$;
(2) If $c_n= \frac{2^{a_n}}{(2^{a_{n+1}})^2+3\times2^{a_{n+1}}+2}$, find the sum $T_n$ of the first $n$ terms of the sequence $c_n$.
(3) If $c_n= \frac{a^{a_n}}{(2^{a_{n+1}})^2+3\times2^{a_{n+1}}+2}$ $(a > 0)$, find all values of $a$ that satisfy the following two conditions simultaneously: ① For any positive integer $n$, $T_n < \frac{1}{6}$; ② For any $m\in(0, \frac{1}{6})$, there exists $n_0\in\mathbb{N}^{*}$, such that when $n\geqslant n_0$, $T_n > m$.

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