
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $2c\cos B=2a-b$.

$(1)$ Find the magnitude of angle $C$,

$(2)$ Given that the sequence $\{a_n\}$ is a geometric sequence with common ratio $q=\cos (A+B)$, and $a_3=6\cos C$, find the general formula for the sequence $\{a_n\}$;

$(3)$ Based on the conclusion of $(2)$, let $b=\log_2\left(\frac{3}{a_{2n+3}}\right)$, 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}
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