
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 that all terms of the geometric sequence $\{a_n\}$ are positive, with the first term $a_1=2$, and $S_n$ is the sum of the first $n$ terms. If $5S_1$, $S_3$, and $3S_2$ form an arithmetic sequence,  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $b_n=\log_2a_n$, and $c_n= \frac {1}{b_nb_{n+1}}$, denote the sum of the first $n$ terms of the sequence $\{c_n\}$ as $T_n$. If for $\forall n\in \mathbb{N}^*$, $T_n\leq k(n+4)$ always holds, find the range of the real number $k$.

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