
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√2+√5 and sin A:sin B:sin C=(√2-1):√5:(√2+1), use the formula S=√{(1/4)[c^2a^2-( (c^2+a^2-b^2)/2 )^2]} to find the area of △ABC.

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