
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
Given the function $f(x)= \frac{bx}{ax^{2}+c}$, where $f′(0)=9$, $a > 0$, $b$, $c∈R$, and $b+c=10$.
1. Find the values of $b$ and $c$ and the monotonic intervals of the function $f(x)$.
2. If there is only one $x_{0}$ in the interval $[1,2]$ such that $f(x_{0})≥slant a$, find the value of the real number $a$.

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