
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
Point $P(x_{0},y_{0})(x_{0}\neq \pm a)$ is on the hyperbola $E$: $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$. $M$ and $N$ are the left and right vertices of the hyperbola $E$, respectively. The product of the slopes of lines $PM$ and $PN$ is $\dfrac {1}{5}$.  
$(1)$ Find the eccentricity of the hyperbola;  
$(2)$ A line passing through the right focus of the hyperbola $E$ with a slope of $1$ intersects the hyperbola at points $A$ and $B$. Let $O$ be the origin, and $C$ be a point on the hyperbola such that $\overrightarrow{OC}=\lambda \overrightarrow{OA}+ \overrightarrow{OB}$. Find the value of $\lambda$.

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