
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
A modified parametric graph is defined by the equations
\[
x = \cos t + \frac{t}{2} + \sin(2t),
\]
\[
y = \sin t.
\]
How many times does this new graph intersect itself between $x = 2$ and $x = 40$?

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