
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 $\triangle ABC$, $B(-\frac{3\sqrt{2}}{2}, 0)$, $C(\frac{3\sqrt{2}}{2}, 0)$, its perimeter is $6+3\sqrt{2}$, $O$ is the midpoint of $BC$, $T$ is on the line segment $AO$, such that $\overrightarrow{TA} = -\overrightarrow{TO}$.
(1) Find the equation of the trajectory $E$ of point $T$;
(2) If $M(m, 0) (0 < m < 1)$, $N(n, 0)$ is on the extension of $OC$, the line passing through point $M$ intersects trajectory $E$ at points $P$ and $Q$, and the line $QN$ intersects trajectory $E$ at another point $R$, such that $(\overrightarrow{MP} + \overrightarrow{MR}) \cdot \overrightarrow{PR} = 0$, find the value of $mn$.

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