
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 intercepted by these curves on the line \(y = 2x\).

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