Add `air_private_input` flag to cairo1-run #7609

Job	Run time
Build release binary for comparisons	45s
Build Cairo programs (cairo_bench_programs)	17s
Build the wasm demo	39s
Build Cairo programs (cairo_proof_programs)	23s
Build Cairo programs (cairo_test_programs)	22s
Build Cairo programs (cairo_1_test_contracts)	19s
Build Cairo programs (cairo_2_test_contracts)	18s
Merge Cairo programs cache	1s
Run Lints	2s
Compute memory and execution traces with cairo-lang (cairo_proof_programs, cairo_proof_trace, 1)	1s
Compute memory and execution traces with cairo-lang (cairo_test_programs, cairo_trace, 2)	2s
Run tests	1s
Make sure all builds work	1s
Upload coverage results to codecov.io	1s
Compute memory and execution traces with cairo-vm	1s
Compare memory and execution traces from cairo-lang and cairo-vm	1s
	3m 14s

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