Add `air_public_input` flag to cairo1-run

[fmoletta]

Air Public Input can now be obtained from cairo 1 proof_mode runs

[codecov[bot]]

Codecov Report

Attention: 6 lines in your changes are missing coverage. Please review.

Comparison is base (352e8a8) 97.21% compared to head (5c19b3b) 97.22%.

Files	Patch %	Lines
cairo1-run/src/main.rs	87.23%	6 Missing :warning:

Additional details and impacted files

```diff @@ Coverage Diff @@ ## main #1539 +/- ## ======================================= Coverage 97.21% 97.22% ======================================= Files 91 91 Lines 36990 37048 +58 ======================================= + Hits 35961 36020 +59 + Misses 1029 1028 -1 ```

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Benchmark Results for unmodified programs :rocket:

Command	Mean [s]	Min [s]	Max [s]	Relative
base big_factorial	2.269 ± 0.022	2.243	2.319	1.00
head big_factorial	2.270 ± 0.028	2.243	2.325	1.00 ± 0.02

Command	Mean [s]	Min [s]	Max [s]	Relative
base big_fibonacci	2.241 ± 0.018	2.211	2.274	1.00 ± 0.01
head big_fibonacci	2.237 ± 0.012	2.225	2.267	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base blake2s_integration_benchmark	8.281 ± 0.076	8.181	8.409	1.01 ± 0.01
head blake2s_integration_benchmark	8.215 ± 0.072	8.129	8.319	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base compare_arrays_200000	2.285 ± 0.008	2.271	2.300	1.00
head compare_arrays_200000	2.306 ± 0.016	2.279	2.325	1.01 ± 0.01

Command	Mean [s]	Min [s]	Max [s]	Relative
base dict_integration_benchmark	1.461 ± 0.010	1.451	1.482	1.01 ± 0.01
head dict_integration_benchmark	1.449 ± 0.008	1.434	1.461	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base field_arithmetic_get_square_benchmark	1.296 ± 0.010	1.286	1.320	1.00 ± 0.01
head field_arithmetic_get_square_benchmark	1.296 ± 0.011	1.281	1.312	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base integration_builtins	8.327 ± 0.084	8.204	8.428	1.02 ± 0.01
head integration_builtins	8.201 ± 0.058	8.144	8.331	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base keccak_integration_benchmark	8.531 ± 0.073	8.456	8.697	1.00 ± 0.01
head keccak_integration_benchmark	8.502 ± 0.083	8.412	8.650	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base linear_search	2.352 ± 0.025	2.319	2.386	1.00
head linear_search	2.363 ± 0.013	2.350	2.394	1.00 ± 0.01

Command	Mean [s]	Min [s]	Max [s]	Relative
base math_cmp_and_pow_integration_benchmark	1.632 ± 0.010	1.620	1.645	1.01 ± 0.01
head math_cmp_and_pow_integration_benchmark	1.621 ± 0.017	1.605	1.647	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base math_integration_benchmark	1.475 ± 0.008	1.464	1.488	1.01 ± 0.01
head math_integration_benchmark	1.462 ± 0.006	1.456	1.471	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base memory_integration_benchmark	1.275 ± 0.009	1.262	1.288	1.00
head memory_integration_benchmark	1.276 ± 0.012	1.266	1.306	1.00 ± 0.01

Command	Mean [s]	Min [s]	Max [s]	Relative
base operations_with_data_structures_benchmarks	1.645 ± 0.008	1.634	1.658	1.01 ± 0.01
head operations_with_data_structures_benchmarks	1.629 ± 0.012	1.616	1.654	1.00

Command	Mean [ms]	Min [ms]	Max [ms]	Relative
base pedersen	607.9 ± 3.1	604.8	614.6	1.00
head pedersen	608.8 ± 3.9	604.8	617.4	1.00 ± 0.01

Command	Mean [s]	Min [s]	Max [s]	Relative
base poseidon_integration_benchmark	1.079 ± 0.010	1.070	1.104	1.00
head poseidon_integration_benchmark	1.082 ± 0.003	1.076	1.086	1.00 ± 0.01

Command	Mean [s]	Min [s]	Max [s]	Relative
base secp_integration_benchmark	1.958 ± 0.010	1.945	1.976	1.01 ± 0.01
head secp_integration_benchmark	1.930 ± 0.007	1.920	1.944	1.00

Command	Mean [ms]	Min [ms]	Max [ms]	Relative
base set_integration_benchmark	779.9 ± 1.7	778.0	783.3	1.00 ± 0.01
head set_integration_benchmark	779.3 ± 4.6	774.2	789.7	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base uint256_integration_benchmark	4.634 ± 0.038	4.587	4.689	1.02 ± 0.01
head uint256_integration_benchmark	4.558 ± 0.028	4.526	4.629	1.00