
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

A unit square \(ABCD\) has point \(F\) as the midpoint of side \(AB\). A circle centered at \(F\) passing through point \(C\) intersects the ray \(AB\) at point \(G\). The circle with diameter \(AG\) intersects segment \(BC\) at point \(H\). Let point \(E\) be the quarter-point of the diagonal \(AC\) closer to \(C\). Let the lengths of the segments \(BE\), \(BH\), and the semicircle \(BF\) be \(e\), \(h\), and \(f\) respectively.

Determine the value of the ratio \(\frac{e-f}{h-f}\) to two decimal places.

How many significant digits of \(\pi\) are required to achieve this precision?

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