
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
Where is Tony's cheese? The bottom right corner of a 3x3 grid :0 We’re doing weekly posts of problems until the start of the in-person tournament on April 13th (Sign up **[here](https://www.stanfordmathtournament.com/competitions/smt-2024)**!) Registrations for our synchronous Online tournament are also open **[here](https://www.stanfordmathtournament.com/competitions/smt-2024-online)**. Also if you’re coming in-person and want to get some pre-competition action, submit your solutions to PoTW [here](https://forms.gle/2FpDoHFeVHg2aqWS6), and the top 3 people with the most correct solutions will receive surprise prize at the tournament :)**Easy:** Consider the curves  $x^2 + y^2 = 1$  and  $2x^2 + 2xy + y^2 - 2x -2y = 0$ . These curves intersect at two points, one of which is  $(1, 0)$ . Find the other one. **Hard:** Tony the mouse starts in the top left corner of a  $3x3$  grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center.

Again feel free to reach out to us at [stanford.math.tournament@gmail.com](mailto:stanford.math.tournament@gmail.com) with any questions, or ask them in this thread!

Best,

[The SMT Team](https://www.stanfordmathtournament.com/our-team)


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