
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) Let \(ABCD\) be an isosceles trapezoid with \(AB = CD = 5\), \(BC = 2\), and \(AD = 8\). Find the height of the trapezoid and the length of its diagonals.

(b) For the trapezoid introduced in (a), find the exact value of \(\cos \angle ABC\).

(c) In triangle \(KLM\), let points \(G\) and \(E\) be on segment \(LM\) so that \(\angle MKG = \angle GKE = \angle EKL = \alpha\). Let point \(F\) be on segment \(KL\) so that \(GF\) is parallel to \(KM\). Given that \(KFEG\) is an isosceles trapezoid and that \(\angle KLM = 84^\circ\), determine \(\alpha\).

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