
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 $\triangle ABC$, $A$, $B$, $C$ are the angles opposite to sides $a$, $b$, $c$ respectively. Given $\sin C= \frac{ \sqrt{10}}{4}$.
(1) If $a+b=5$, find the maximum value of the area of $\triangle ABC$.
(2) If $a=2$, $2\sin ^{2}A+\sin A\sin C=\sin ^{2}C$, find the length of $b$ and $c$.

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