
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
Elective 4-4; Coordinate System and Parametric Equations

In the plane Cartesian coordinate system $xOy$, a line $l$ passes through the point $P(-1,-2)$ with its parametric equations given by $\begin{cases} & x=-1+t\cos 45^\circ, \\ & y=-2+t\sin 45^\circ \\ \end{cases}$ (where $t$ is the parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho\sin \theta\tan \theta=2a(a > 0)$. The line $l$ intersects the curve $C$ at two distinct points $M$ and $N$.

(Ⅰ) Find the standard equations of curve $C$ and line $l$;

(Ⅱ) If $|PM|=|MN|$, find the value of the real number $a$.

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