
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
(a) Find the value of the real number  $k$ , for which the polynomial  $P(x)=x^3-kx+2$  has the number  $2$  as a root. In addition, for the value of  $k$  you will find, write this polynomial as the product of two polynomials with integer coefficients.
(b) If the positive real numbers  $a,b$  satisfy the equation   $$ 2a+b+\frac{4}{ab}=10, $$   find the maximum possible value of  $a$ .

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