
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 an isosceles trapezoid \(ABCD\) with lateral sides \(AB\) and \(CD\) both equal to 10, perpendiculars \(BH\) and \(DK\) are drawn from vertices \(B\) and \(D\) to the diagonal \(AC\). It is known that the bases of the perpendiculars lie on segment \(AC\) and \(AH : AK : AC = 5 : 14 : 15\). Find the area of trapezoid \(ABCD\).

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