
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 a convex quadrilateral \(ABCD\), the diagonals \(AC\) and \(BD\) are \(a\) and \(b\) respectively. Points \(E, F, G,\) and \(H\) are the midpoints of sides \(AB, BC, CD,\) and \(DA\) respectively. The area of quadrilateral \(EFGH\) is \(S\). Find the diagonals \(EG\) and \(HF\) of quadrilateral \(EFGH\).

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