
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
For the ellipse $$C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$$, its foci are denoted as F<sub>1</sub> and F<sub>2</sub> respectively. Point P is on ellipse C, and it satisfies $$\overrightarrow{PF_{1}} \cdot \overrightarrow{F_{1}F_{2}} = 0, |\overrightarrow{PF_{1}}| = \frac{\sqrt{5}}{5}, |\overrightarrow{PF_{2}}|= \frac{9\sqrt{5}}{5}$$.  
(1) Determine the equation of ellipse C.  
(2) Suppose O is the origin, and a moving line l passes through focus F<sub>1</sub> and intersects ellipse C at points M and N. Is there a constant t such that $$\overrightarrow{OM} \cdot \overrightarrow{ON} + t \overrightarrow{F_{1}M} \cdot \overrightarrow{F_{1}N}$$ is a constant? If it exists, find the value of t, otherwise explain why not.

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