Let f_m denote the number of sequence triples ({x₀, x₁, $\dots$, x_m}, {y₀, y₁, $\dots$, y_m}, {z₀, z₁, $\dots$, z_m}) such that:

$1\cdot$ x₀ = y₀ = z₀ = 0

$2\cdot$ Sequences x_i, y_i, z_i are all strictly increasing

$3\cdot$ x_m + y_m + z_m = 4040

Then calculate $\bigl($ $\sum_{i=1}^{\infty}$ f_i $\bigr)$ mod $10^{9}+7$.