
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
In the Cartesian coordinate system xOy, with O as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of curve C<sub>1</sub> is $\rho^2(3+\sin^2\theta)=12$, and the parametric equation of curve C<sub>2</sub> is $$\begin{cases} x=1+t\cos\alpha \\ y=t\sin\alpha \end{cases}$$ (t is the parameter), $$\alpha\in(0, \frac{\pi}{2})$$.
(1) Find the rectangular coordinate equation of curve C<sub>1</sub> and determine what type of curve it is;
(2) Suppose the intersection points of curve C<sub>2</sub> and curve C<sub>1</sub> are A and B. When $$|PA|+|PB|=\frac{7}{2}$$, find the value of $\cos\alpha$.

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