
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 $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a=2\sqrt{2}$, $b=5$, and $c=\sqrt{13}$. Find:

$(Ⅰ)$ The measure of angle $C$;

$(Ⅱ)$ The value of $\sin A$;

$(Ⅲ)$ The value of $\sin \left(2A+\frac{\pi}{4}\right)$.

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