= Solution
For a 1 micrometre laser, $n_c\simeq1.1\times10^{27}$ m$^{-3}$ and a cold underdense <wave> has $\gamma_\phi^2\simeq n_c/n_e$. At peak <wave> scale $E_0$, optimal passage through an accelerating half-cycle gives ideal stage gain $\Delta\mathcal E\sim4(n_c/n_e)m_ec^2$; crest-to-zero injection gives half that. With the illustrative $n_e\simeq7\times10^{22}$ m$^{-3}$, the optimal ideal gain is about 32 GeV and a 9.9 TeV increment needs order 300 stages, or order 600 with the crest convention.
\b[TODO — no unique stage count follows without the missing <density> and laser/wake <amplitude> assumptions.] Depletion, diffraction, guiding, beam loading, injection, <energy> spread, driver evolution and interstage losses can all increase or otherwise alter the requirement. The parametric count is $N\gtrsim\Delta\mathcal E_{\mathrm{total}}/[4\eta(n_c/n_e)m_ec^2]$, with the phase and field conventions explicitly retained.
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