
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 a quadratic function $f(x)=ax^2+bx+c$ ($a\neq 0$) that satisfies the following three conditions:

1) The graph of the function $f(x)$ passes through the origin;
2) The equation of the axis of symmetry of the function $f(x)$ is $x=-\frac{1}{2}$;
3) The equation $f(x)=x$ has two equal real roots.

Let $g(x)=f(x)-(1+2\lambda)x$.

(1) Find the analytic expression of the function $f(x)$;
(2) Find the range of real numbers $\lambda$ that make the inequality $g(x)\geqslant -1$ always true;
(3) Given that the minimum value of the function $g(x)$ on $[-2,1]$ is $-3$, find the value of the real number $\lambda$.

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