In rectangle \(ABCD\), \(AD=1\), \(P\) is on \(\overline{AB}\), and \(\overline{DB}\) and \(\overline{DP}\) trisect \(\angle ADC\). The perimeter of \(\triangle BDP\) is in the form \(k+\frac{m\sqrt{n}}{3}\), where \(k\), \(m\), and \(n\) are integers. Find the value of \(k + m + n\).

Let's think step by step and output the final answer within \boxed{}.
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