
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 a triangle ABC with a perimeter of $2 \sqrt {2}+ \sqrt {5}$ and $\sin A : \sin B : \sin C = (\sqrt {2}-1) : \sqrt {5} : (\sqrt {2}+1)$, use the formula $S= \sqrt { \frac {1}{4}[c^{2}a^{2}-( \frac {c^{2}+a^{2}-b^{2}}{2})^{2}]}$ to find the area of triangle ABC.

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