
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
The leader of the patriots, learning from the example of Mátyás Sándor, sent an equation to his friends:

$$
2 x^{2}+4 x y+7 y^{2}-12 x-2 y+N=0
$$

The secret message was that the uprising should take place on the $N$-th day of the month. The patriots often said that the uprising was the only solution. This also characterized $N$: the equation has a unique real solution $(x, y)$. What is the value of $N$?

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