
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 Systems and Parametric Equations)
In the rectangular coordinate system, the origin $O$ is used as the pole and the positive semi-axis of the $x$-axis is used as the polar axis to establish a polar coordinate system. The polar coordinates of point $P$ are $(1, \pi)$, the curve $C$ is defined as $\rho = 2\sqrt{2}a \sin\left(\theta + \frac{\pi}{4}\right) \left(a > 0\right)$, and the line $l$ passes through point $P$ with the parametric equations: $\begin{cases} x = m + \frac{1}{2}t \\ y = \frac{\sqrt{3}}{2}t \end{cases} (t \text{ is a parameter})$. The line $l$ intersects the curve $C$ at points $M$ and $N$.

(I) Write the rectangular coordinate equation of curve $C$ and the standard form equation of line $l$.

(II) If $|PM| + |PN| = 5$, find the value of $a$.

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