
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
Given the family of curves
$$
2(2 \sin \theta - \cos \theta + 3) x^{2} - (8 \sin \theta + \cos \theta + 1) y = 0,
$$
where $\theta$ is a parameter. Find the maximum length of the chord that these curves cut on the line $y = 2 x$.

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