
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Determine the sum of all prime numbers $p$ for which there exists no integer solution to the system of congruences:
1. $30(6x+1) \equiv 5 \pmod p$
2. $7x + 2 \equiv 0 \pmod p$

Present the answer in LaTex format: \boxed{Your answer}
user