Add `air_private_input` flag to cairo1-run #7610

Job	Run time
Build release binary for comparisons	14s
Build Cairo programs (cairo_bench_programs)	12s
Build Cairo programs (cairo_proof_programs)	16s
Build the wasm demo	12s
Build Cairo programs (cairo_test_programs)	15s
Build Cairo programs (cairo_1_test_contracts)	13s
Build Cairo programs (cairo_2_test_contracts)	12s
Run Lints	1s
Merge Cairo programs cache	1s
Compute memory and execution traces with cairo-lang	1s
Compute memory and execution traces with cairo-vm	1s
Make sure all builds work	1s
Compare memory and execution traces from cairo-lang and cairo-vm	1s
Run tests	1s
Upload coverage results to codecov.io	1s
	1m 42s

Provide feedback