
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
For all real numbers $x$ and $y$, define the mathematical operation $\#$ such that the following conditions apply: $x\ \#\ 0 = x, x\ \#\ y = y\ \#\ x$, and $(x + 1)\ \#\ y = (x\ \#\ y) + 2y + 1$. Calculate the value of $7\ \#\ 3$.

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