
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 arithmetic sequence $\{a_n\}$ with a common difference not equal to $0$, the sum of the first $n$ terms is $S_n$, satisfying $S_3=a_4+4$, and $a_2, a_6, a_{18}$ form a geometric sequence.

$(1)$ Find the general formula for the sequence $\{a_n\}$;

$(2)$ Let $b_n=\frac{a_n}{2^n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$;

$(3)$ Let $c_n=\sqrt{S_n+t}$, if $\{c_n\}$ is an arithmetic sequence, find the value of the real number $t$.

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