
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 $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $2\sin^2\left(\frac{A+C}{2}\right)+\cos 2B=1$  
(1) If $b=\sqrt{13}$ and $a=3$, find the value of $c$;  
(2) Let $t=\sin A \sin C$, find the value of $A$ when $t$ reaches its maximum value.

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